Rank Efficiency: Investigating a Widespread
Ordinal Welfare Criterion
Clayton R. Featherstone∗
December 10, 2013
UNDER REVISION: COMMENTS WELCOME
Abstract
Many institutions that allocate scarce goods based on rank-order preferencesgauge the success of their assignments by the resulting rank distributions, thatis, how many participants get their first choice, how many get their secondchoice, and so on. For example, San Francisco Unified School District, Teachfor America, and Harvard Business School all evaluate assignments in this way.Preferences over rank distributions capture the practical (but non-Paretian)intuition that hurting one agent to help others might be desirable. Motivatedby this, call an assignment rank efficient if its rank distribution cannot feasiblybe stochastically dominated. Rank efficient mechanisms are simple linear pro-grams that can either be solved all at once by a computer or through an intuitivesequential improvement process where at each step, the policy-maker executesa potentially non-Pareto-improving trade cycle. Both methods are used in thefield. Rank efficiency also dovetails nicely with previous literature: it is a re-finement of ordinal efficiency (and hence of ex post efficiency). Although rankefficiency is theoretically incompatible with strategy-proofness, rank efficientmechanisms can admit a truth-telling equilibrium in low information environ-ments. Preference data from Featherstone and Roth (2011) show that if agentswere to truthfully reveal their preferences, a rank efficient mechanism couldsignificantly outperform commonly considered alternatives like random serialdictatorship and the probabilistic serial mechanism. Finally, a competitiveequilibrium mechanism like that of Hylland and Zeckhauser (1979) generatesa straightforward generalization of rank efficiency and sheds light on how rankefficiency interfaces with fairness considerations.
∗The Wharton School, University of Pennsylvania
1
1 Introduction
Mechanisms that map submitted ordinal preferences into an assignment of agents to
objects are common, and institutions that use them often gauge success by looking
at rank distributions, that is, at how many participants get their first choice, how
many get their second choice, and so on. For example, when school districts announce
the results of a choice-based student placement system, they report things like how
many students were assigned to one of their top three choices and how many were
unmatched (NYC Department of Education 2009, San Francisco Unified School Dis-
trict 2011).1 Another example is the mechanism used by Teach for America to match
teachers to regions, which explicitly selects an assignment based on rank distribu-
tion.2 And finally, consider the strategy-proof random serial dictatorship used by
Harvard Business School to match MBAs to the country in which they fulfill their
global immersion requirement.3 In the first year the mechanism was run, administra-
tors were so concerned about the number of students who, in spite of ranking it last,
were assigned to an unpopular country, that they seriously considered exchanging
those students with others who had only given the unpopular country a moderately
low rank. After being reminded that such rearrangement would violate a promise of
strategy-proofness, they reluctantly stayed with the original assignment. Indeed, a
big part of strategy-proofness is making it safe for agents to reveal that they somewhat
like objects that most other agents strongly dislike.
This last anecdote is not surprising in light of the fact that random serial dicta-
torship is merely Pareto efficient. It could easily yield an assignment where shifting
one student from first to second choice could enable four other students to shift from
second to first choice.4 While this sort of trade seems good for overall welfare, Pareto-
agnosticism only insists on trades that weakly improve all agents. Motivated by this
common-sense intuition and encouraged by the widespread use of rank distribution-
based objectives in the field, define a feasible assignment to be rank efficient if its
rank distribution cannot be first-order stochastically dominated by that of another
1SFUSD also made frequent use of rank distributions to help them understand how to comparedifferent school choice mechanisms proposed by myself, Atila Abdulkadiroglu, Muriel Niederle, ParagPathak, and Al Roth over the course of the 2009-2010 school year in which we advised them.
2Teach for America is a nationwide non-profit that sends mostly new college graduates to teachin at-risk public schools. Al Roth and I are actively involved with helping the organization tostreamline its assignment process. See Section 4.4 for more.
3This paper discusses the HBS global immersion match in Section 7. Al Roth and I are activelyinvolved with the design of the match. See Featherstone and Roth (2011).
4Consider the following five agent, five position example, where the circles and boxes representtwo different assignments. A policy-maker looking at the boxed assignment might be tempted to
2
feasible assignment. Rank efficiency formalizes the intuition that, if there is no reason
to differentiate between one agent getting his kth choice or another, then it can be
welfare improving to hurt one agent in order to help others.
Mechanisms that implement rank efficient assignments can also be characterized
in terms of this intuition. Consider giving values to the different ranks, such that
the value of a first choice is more than that of a second choice, and so on. To see if
a potentially non-Pareto-improving trade cycle should be executed, simply calculate
whether doing so will increase the overall “rank-value” of the market. For instance,
when considering a two-person trade that moves one agent from his first to his third
choice, and the other from his fourth to his second, the relevant change in rank-value
would be (v3−v1)+(v2−v4), which could be positive or negative, depending on what
v was chosen. I show that an assignment is rank efficient if and only if there is some
valuation vector v such that all possible trade cycles yield a non-positive value. In
other words, starting with some assignment and sequentially improving it by executing
trade cycles that are rank-value improving, but potentially not Pareto-improving, will
eventually yield a rank efficient assignment. Teach for America used exactly this kind
of process prior to Spring 2011. Such a process can also be easily automated with a
simple linear program, and in fact, linear programming mechanisms, although not as
prevalent as random serial dictatorships, can be found in the field: they are used to
match house-officers5 to hospitals in Cambridge and London (Roth 1991), and since
Spring 2011, by Teach for America as well.6
Rank efficiency also dovetails nicely with two efficiency concepts that have domi-
nated the theoretical literature on assignment markets. Ex post efficiency requires
that there be no other deterministic assignment that improves all agents. Stepping
back to lotteries over assignments (that is, to the interim stage), and thinking about
agents as possessing bundles of object probability shares, ordinal efficiency (Bogo-
implement a trade cycle to get the circled assignment instead.
1 : a e©2 : a© b
3 : b© c
4 : c© d
5 : d© e
5Roughly speaking, house-officers are a U.K. analog to medical residents in the American market(Roth 1984).
6One might object that Al and I suggested this mechanism; however, in Section 4.4, I argue thatTeach for America adopted a linear programming assignment method, not because it was somethingnew, but because it duplicated the non-automated process of the past in much less time.
3
molnaia and Moulin 2001) requires that there be no other assignment of shares that
stochastically improves all agents. Rank efficiency is a refinement of both concepts.7
Looking to mechanisms, random serial dictatorship8 is ex post efficient, strategy-
proof, and ubiquitous in the field, while the probabilistic serial mechanism9 is or-
dinally efficient, generically non-strategy-proof, and absent in the field. Given the
theoretical appeal of ordinal efficiency, it was unclear why there are no ordinally ef-
ficient mechanisms in the field. Since rank efficient mechanisms are also ordinally
efficient, however, their presence resolves this institutional puzzle.
In the field, the conflict between strategy-proofness and refined efficiency is framed
as a decision between random serial dictatorship and linear programming. How does
the market designer choose which mechanism is the right one? Do rank efficient mech-
anisms ever make sense, or are they “mistakes” made by uninformed policy-makers?
Two exercises inform these concerns. First, in low information environments, like
those of Roth and Rothblum (1999), rank efficient mechanisms admit a truth-telling
equilibrium. This is how Teach for America justifies using a potentially manipulable
linear programming mechanism: teachers only apply once, are geographically sepa-
rated, and know little about what regions are popular and how the mechanism is run.
The second exercise uses empirical data (Featherstone and Roth 2011) from Harvard
Business School’s strategy-proof global immersion match. Under truth-telling, a rank
efficient mechanism yields more than a 15% improvement in the number of MBAs
who are assigned to a first or second choice country (out of 11) when compared to the
assignment given by random serial dictatorship. Although these gains might be un-
dermined by strategic preference manipulation, the exercise shows that in a naturally
occurring environment, the costs of strategy-proofness can be quite large. Together,
these exercises indicate that in certain environments, it would be a mistake to not at
least consider using a rank efficient mechanism.
Finally, the paper considers a class of ordinal mechanisms based on Hylland and
Zeckhauser (1979). These mechanisms assume cardinal preferences to rationalize the
submitted ordinal preferences, give each agent a budget of artificial10 money, and
7Given the limited information of an ordinal setting, one can interpret rank efficiency as ex anteefficiency if the v vector from the previous paragraph is thought of as an assumption about thepolicy-maker’s beliefs.
8That is, draw an ordering of the agents from some fixed distribution over orderings, and letthem choose their objects in that order. See Section 3.
9The probabilistic serial mechanism is the baseline ordinally efficient mechanism Bogomolnaiaand Moulin (2001). See Appendix C for a formal definition.
10That is, the money only exists within the mechanism and is not intrinsically valued by theagents. Its only worth is to purchased probability shares within the mechanism.
4
then finds prices such that agents would buy optimal allocations of object probability
shares that clear the market (in other words, calculates a competitive equilibrium
assignment). Such mechanisms always yield ordinally efficient assignments, and in
fact, they yield a refinement of ordinal efficiency that generalizes rank efficiency by
allowing different agents to be weighted differently when the rank distribution is tab-
ulated. Hence, there is a mapping between generalized rank efficient mechanisms11
and competitive equilibrium mechanisms. Perhaps surprisingly, a competitive equi-
librium where all agents have the same budget does not correspond to an assignment
supported by the rank efficient mechanism that places equal weights on each agent.
In this sense, procedural fairness means different things under the two types of mech-
anisms. Equal budgets in a competitive equilibrium mechanism correspond to justice
based on envy-freeness (Varian 1974, Dworkin 1981), while equal agent weights in a
rank efficient mechanism correspond to justice based on a utilitarian12 version of the
Rawlsian veil-of-ignorance (Harsanyi 1975).
The rest of the paper is organized into three parts. The first part (Sections 2-6)
presents rank efficiency and rank efficient mechanisms. It then relates these mech-
anisms to how Teach for America assigns teachers and to the broader theory liter-
ature concerning efficiency in assignment markets. The second part (Sections 7-8)
argues that rank efficient mechanisms could potentially yield large welfare gains by
demonstrating two things. First, preference data from the strategy-proof HBS global
immersion match demonstrates that if a rank efficient mechanism could access the
true preferences of the agents, it could yield big efficiency gains. Second, although
strategy-proofness and rank efficiency are theoretically incompatible, rank efficient
mechanisms can admit a truth-telling equilibrium in low information environments.
The third part (Sections 9-10) generalizes rank efficiency and shows that the gen-
eralization is closely related to the competitive equilibrium mechanisms of Hylland
and Zeckhauser (1979). This result can be leveraged to interpret the theoretical in-
compatibility of rank efficiency and envy-freeness as a wedge between two important
concepts of justice. The conclusion discusses how rank efficient mechanisms should
enter into the discussion about the costs of strategy-proofness, and how they raise
interesting questions about the costs of Pareto-agnosticism.
11As just discussed, the rank efficient mechanism gives an assignment vk points for assigning anyagent his kth choice. The generalized rank efficient mechanism weights agents according to somevector (α)a: the value of assigning an agent a to his kth choice is worth αa · vk.
12That is, maximizing expected utility from behind the veil. If one were to follow Rawls (1972),then the objective would be maximin, which would lead to a concept of justice that more closelyresembles that of Dworkin.
5
2 The model
Consider assigning each agent a from set A to exactly one object o from set O.
Further, let there be qo copies of each object. Sometimes the set of objects will
include a special “null” object, ∅, that denotes an agent’s outside option; for that
reason, it is modeled as a non-scarce good, that is, q∅ = |A|. When agents have no
outside option,13 the null object is omitted. Whether there is a null object or not, I
will require that there are enough objects that every agent can be feasibly matched,
that is∑
o∈O qo ≥ |A|.A deterministic assignment is a function that maps agents to objects feasi-
bly, that is, each agent is only mapped to one object, and no more than qo agents are
assigned to any given o ∈ O. Such an assignment can be represented as an |A| × |O|matrix x where xao ∈ 0, 1,
∑o xao = 1, and
∑a xao ≤ qo, for all a ∈ A and o ∈ O.
xao = 1 means that agent a is assigned to object o; xao = 0 means that agent a is
not. A random assignment is a lottery over deterministic assignments, which can
be represented as the corresponding convex combination over deterministic assign-
ment matrices. As such, random assignment matrices will have a similar structure
to deterministic assignment matrices, except xao ∈ [0, 1]. By the extension of the
Birkhoff-von Neumann theorem (Birkhoff 1946) put forth in Budish et al. (2011), any
such matrix represents some lottery over deterministic assignments.14 Call this lottery
a lottery representation of the random assignment matrix, and the deterministic
assignments the representation’s support. Most of the time, this paper will use the
freedom afforded by the Budish et al. theorem to focus on matrix representations.
Moving to the individual agent, call the ath row of a random assignment x agent
a’s allocation, xa. Each agent a is endowed with an ordinal preference %a over
O; note that indifferences are allowed. It will often prove convenient to express
agents’ preferences in terms of rank functions, ra(·), which are mappings from O to
1, . . . , |O|+ 1.15 When preferences are strict, there are several equivalent ways to
define the standard rank function: ra(o) = |o′ ∈ O|o′ %a o|, or ra(o) = |o′ ∈ O|o′ a o|+
13The obvious example is military postings. Perhaps a more subtle example is public schoolassignment. In San Francisco, students are not required to rank all schools, but if they cannot beassigned to a school they ranked, they are generally given an administrative assignment. In thissort of situation, failing to rank all schools is equivalent to the student saying to the school district,“beyond what I ranked, you can fill out the rest of my rank-order list for me.”
14This lottery need not be unique. Generally, this is an unimportant detail, as all such lotteriesover assignments induce the same lottery over objects for each agent, but sometimes the subtlety isimportant. See Remark 1.
15|O|+ 1 is included for technical reasons that will become clear in the next paragraph.
6
1, or even ra(o) = |[o′] ∈ O/ ∼a |o′ %a o|.16
With indifferences, however, these definitions are no longer equivalent. Consider
%a = a b ∼ c d. The ranks of (a, b, c, d) under these definitions are, respectively,
(1, 3, 3, 4), (1, 2, 2, 4), and (1, 2, 2, 3). Another ambiguity in how to think of the rank
function concerns the null object, ∅. One could of course treat ∅ just like any other
object; however, policy-makers often report the number of unassigned agents as a
separate category that is, in some sense, worse than any other rank.17 In the context
of the mechanism to be introduced in this paper, one can model this by setting
ra(∅) = |O| and for all o ≺a ∅, ra(o) = |O| + 1.18 In Section 8, I show that the
specifics of the mapping from %a to ra(·) (the ranking scheme) can affect incentives
for truth-telling. For now, the theory can be built around any of these definitions so
long as o′ a o ⇔ ra(o′) < ra(o) and o′ ∼a o ⇔ ra(o
′) = ra(o).19 Finally, define an
ordinal assignment mechanism to be a mapping from submitted preferences to
random assignments.
3 Ex post efficiency
Before introducing rank efficiency, I introduce the baseline concept of efficiency for
assignment markets, ex post efficiency. Informally, a “good” random assignment is
a lottery such that, regardless of which deterministic assignment is realized, there
won’t be a rearrangement of the objects that all agents weakly prefer (strictly for
one). Formally,
Definition 1. A feasible deterministic assignment x is said to ex post dominate
another deterministic assignment x if xa %a xa for all a ∈ A, and there is some a where
the preference is strict. A feasible deterministic assignment is ex post efficient if
it is not ex post dominated. A random assignment x is ex post efficient if it is a
lottery over ex post efficient deterministic assignments.
Remark 1. Although non-intuitive, it is possible for the matrix representation of an
ex post efficient random assignment to have a lottery representation whose support
16O/ ∼a is the set of indifference classes of O with respect to ∼a; hence, |[o′] ∈ O/ ∼a |o′ %a o|is the number of indifference classes whose objects are weakly preferred to o. When preferences arestrict, the indifference classes are all singletons.
17This can be clearly seen in the press releases from the school matches in San Francisco andNew York City (NYC Department of Education 2009, San Francisco Unified School District 2011).
18The rank-value mechanisms (see Section 4.2) introduced in this paper effectively ignore agents’preferences below ∅.
19This requirement can be relaxed for objects ranked below ∅ under mechanisms that do notconsider the rankings of unacceptable objects.
7
is not entirely ex post efficient (see Abdulkadiroglu and Sonmez (2003a) for an ex-
ample). This is one instance where the non-uniqueness of the Birkhoff-von Neumann
decomposition requires careful handling. Matrix representations of rank efficient (and
ordinal efficient) random assignments do not have this problem.
Remark 2. The definition of ex post efficiency for random assignments can be equiv-
alently expressed in terms of a state-by-state domination concept. See the Section A
in the Appendix for more.
To characterize the mechanisms that implement ex post efficiency, it is helpful
to formally define serial dictatorships, which, informally, allow the agents to choose
their objects in some predefined order.
Definition 2. Serial dictatorship maps the reported strict preferences of the
agents, (a), and a permutation π of (1, . . . , |A|) to a deterministic assignment ac-
cording to to the recursion relations O0 = O, xaπ(k) = maxaπ(k) Ok−1, and Ok =
Ok−1 \xaπ(k) . Denote the assignment that results from these recursions as SD[; π].20
Random serial dictatorships simply randomize the ordering used in the serial
dictatorship.
Definition 3. Random serial dictatorship maps the reported strict preferences
of the agents, (a), and a distribution over permutations of (1, . . . , |A|), p(π), to the
random assignment that results from a lottery that picks SD[; π] with probability
p(π). Denote the resulting random assignment as RSD[; p].
Random serial dictatorships are fundamentally related to the class of ex post effi-
cient assignments: they always generate such assignments, and any such assignment
can be generated by some random serial dictatorship.
Proposition 1. An assignment x is ex post efficient relative to strict preferences if and only if there exists a distribution over permutations p such that x = RSD[; p]
Proof. Bogomolnaia and Moulin (2001) show that a deterministic assignment x is ex
post efficient if and only if there exists an ordering π such that x = SD[; π]. The
“if” part of the preceding Proposition follows because RSD[; p] is a lottery over ex
post efficient assignments, by construction. The “only if” part follows because the
lottery representation of x whose support is ex post efficient gives us the distribution
over orderings required.
20Abusing notation, for deterministic assignments, let xa denote both the ath row of x and theobject that xa represents.
8
Remark 3. Random serial dictatorships with indifferences have been discussed in the
literature (Svensson 1994, 1999, Bogomolnaia, Deb and Ehlers 2005), albeit from
more theoretical perspective. For the sake of simplicity, I have relegated discussion
of how indifferences can be practically incorporated into the framework of random
serial dictatorship to Section B of the Appendix.
Random serial dictatorship is also strategy-proof.
Proposition 2 (Roth and Sotomayor 1992). If the distribution over orderings does
not depend on the submitted preferences of the agents, then random serial dictatorship
is strategy-proof.
Strategy-proofness, ex post efficiency, and the simplicity of the algorithm are
strong reasons to expect the random serial dictatorships to be widely present in the
field, which is indeed the case.
4 Rank efficiency and the rank-value mechanisms
Many policy-makers gauge their success by looking to the rank distribution of the
assignment they choose, that is, at how many agents get their first choice, how many
get their second choice, and so on. Such a heuristic is a natural way to deal with a dif-
ficulty of Pareto-agnosticism: it can overlook potentially welfare improving trade-offs
in which one agent is hurt but others are helped. In this section, I formally intro-
duce rank efficiency and show that it is equivalent to a specific method of determining
whether a trade cycle is welfare improving. This method can be operationalized either
by simple linear programs or by a sequential improvement process.
4.1 Rank efficiency
Consider the cumulative frequency distribution of ranks received by the agents in a
market. Formally, define the rank distribution of assignment x to be
Nx(k) ≡∑a∈A
∑o∈O
1ra(o)≤k · xao
Nx(k) is the expected number of agents who get their kth choice or better under
assignment x.
Definition 4. A random assignment x is rank-dominated by a feasible assignment
x if the rank distribution of x first-order stochastically dominates that of x, that is,
9
N x(k) ≥ Nx(k) for all k (strict for some k). A feasible random assignment is called
rank efficient if it is not rank-dominated by any other feasible assignment.
In fact, rank efficiency is a refinement of ex post efficiency. Formally,
Proposition 3. If x is rank efficient, then x is ex post efficient; however, the converse
need not hold.
Proof. The example from Footnote 4 shows that the converse need not hold, as the
circles assignment rank dominates the boxes assignment, even though both are ex
post efficient. For the forward implication, by way of contradiction, say that the
support of x contains a deterministic assignment ξ that is ex post dominated by ξ.
It must be that N ξ(k) ≥ N ξ(k),∀k (strict for some k), since ξ weakly improves the
allocation of all agents (strict for one). Let x be the random assignment in which
ξ is replaced with ξ in the lottery representation of x. Now, by linearity, the rank
distribution of a convex combination of assignments is just the corresponding convex
combination of the rank distributions of those assignments. Hence, it must be that
N x(k) ≥ Nx(k),∀k (strict for some k), a contradiction.
Note that it follows directly from the proof that, unlike with ex post efficient
assignments, any lottery representation of a matrix representation of a rank efficient
assignment must have a completely rank efficient support.
Claim 1. The support of any lottery representation of a random assignment ma-
trix that represents a rank efficient assignment must consist entirely of rank efficient
deterministic assignments.
The example in Figure 1a helps to make the concept of rank efficiency more con-
crete. Define x© and y to be the assignments represented by the circles and boxes.
Also consider the output of the uniform random serial dictatorship (all serial dicta-
torship orderings are equally likely), xU−RSD.21 The rank distributions of the three
assignments are listed in Figure 1c. x© rank dominates the others since, column-by-
column, it has a larger rank distribution. Hence, this example shows that random
serial dictatorship can yield rank dominated assignments. Of course, this will not
always be the case (rank efficiency only guarantees that no other assignment rank
dominates); in this example, it is due to the following claim, whose proof illustrates
how to show that an assignment is the unique rank efficient assignment.
21In Section C of the Appendix, I will show that the assignment that results from the probabilisticserial mechanism (introduced in Section 6) is also rank-dominated by x©.
10
1 : a c© d
2 : a© b d
3 : b© c d
(a) Preferences
xU -RSD
a b c d
1 : 1/2 0 1/2 0
2 : 1/2 1/6 0 1/3
3 : 0 5/6 1/6 0
(b) Random assignments
Nx(1) Nx(2) Nx(3)
x© 2 3 3
y 1 3 3
xU -RSD 11/6 8/3 3
(c) Rank distributions
Figure 1: A concrete example
Claim 2. In the simplified example, x© is the unique rank efficient assignment.22
Proof. Claim 1 means that it is sufficient to show that x© is the unique deterministic
rank efficient assignment. The three agents’ first choices only cover two objects;
hence, Nx(1) ≤ 2, and since there are only three agents, Nx(2) ≤ 3. x© hits this
upper bound. Any other assignment that attains the bound must give b to 3 and split
a between 1 and 2. Since b was given to 3, if 2 gets any a, Nx(2) will be strictly less
than 3. Thus, x© uniquely attains the bound, demonstrating that it is the unique
rank efficient assignment.
A corollary of the previous claim is that any rank efficient mechanism will choose
x©. The natural next step is to look for a method of calculating the rank efficient
assignments for a more general setting.
4.2 The rank-value mechanisms
Begin by defining a valuation to be a sequence (vk)|O|+1k=1 of strictly positive real
numbers such that vk > vk+1 for all k ∈ 1, · · · , |O|. vk will be, in some sense, the
22One might argue that this fact casts doubt on rank efficiency as a concept. What if I hadgood reason to care more about 1 than 2 and 3? Section 9.1 will define a generalized version ofrank efficiency in which agents can be weighted differently when the rank distribution is tabulated.Everything shown thus far has an analog to a world in which the weighting over students is fixed,but not uniform.
11
“value” that the rank-value mechanism places on kth choice allocations, so there is no
need for more dimensions than |O|.23 Also note that the strictness of the inequality
is essential; kth choice allocations must be strictly more valuable than (k+1)th choice
allocations.
Definition 5. The rank-value mechanism with respect to valuation v (or the
v-rank-value mechanism, for short) maps agent rank orderings to a maximizer of
the following linear program.
maxx
∑a∈A
∑o∈O
vra(o) · xao
s.t.∑a
xao ≤ qo, ∀o ∈ O∑o
xao ≤ 1 , ∀a ∈ A
xao ≥ 0 ,∀o
∀a
∈ O
∈ A
Assignments in the arg max are said to be supported by the v-rank-value mech-
anism.
This mechanism gives a score to every possible agent-object pair and then finds a
feasible assignment that maximizes the sum of those scores. Giving a kth choice to
an agent is worth vk, regardless of which agent gets it. The mechanism can also be
interpreted as making an assumption about agent’s cardinal utilities and maximizing
a welfare sum. See Section 5 for more on interpreting the rank-value mechanism.
Finally, note that the linear program for the rank-value mechanism can be easily
computed.24 Now, I turn to the properties of the mechanism.
Claim 3. For any assignment x in the arg max,∑
o xao = 1 for all a ∈ A.
Proof. All elements of the valuation vector are strictly positive by definition, so all
agents will be assigned as many probability shares as possible.
23See the end of Section 2 for a reminder about why one might want an (|O|+ 1)th rank.24Many polynomial time algorithms for this program exist, such as the Lawler (1976) improvement
of the Hungarian algorithm (Egervary 1931, Kuhn 1955, Munkres 1957), which has computational
complexity O(
(∑
o qo)3)
(Burkard, Dell’Amico and Martello 2009). Also, the simplex algorithm
can be used. Although it is worst-case exponential, smoothed analysis indicates that parametersthat would force the simplex to the exponential worst case are, in some sense, of measure zero, thatis, the simplex method has polynomial smoothed complexity (Spielman and Teng 2004). In the HBSand Teach for America applications mentioned in this paper, I used CPLEX’s implementation of thesimplex algorithm; solve times were less than a minute.
12
Having the unit-demand constraint be an inequality instead of an equality allows
the use of duality theorems later on, but the difference is not substantive. A rank-
value mechanism will also never assign an agent to something he ranked lower than
∅, that is, it is individually rational.
Claim 4. The v-rank-value mechanism is individually rational.
Proof. Suppose there is an assignment in the arg max that assigns agent a to some-
thing he likes less than ∅. Then, moving a to ∅ increases the objective and is feasible,
since vk > vk+1 and q∅ = |A|.
Finally, the v-rank-value mechanisms won’t allow objects to be unassigned if
they could be used to improve an agent’s welfare. Formally, an assignment is non-
wasteful if xao > 0 ⇒ [o′ a o⇒∑
a xao′ = qo′ ]. A mechanism is non-wasteful if it
always chooses a non-wasteful assignment.
Claim 5. The v-rank-value mechanism is non-wasteful.
Proof. By way of contradiction, assume otherwise. It is feasible for an agent to
swap any shares he has with unclaimed shares, and doing so increases the objective,
contradicting optimality.
Finally, although knowing which maximizer of the linear program is chosen is
not always necessary for the theory, the selection is important for implementation.
Consider the set of deterministic assignments. Since this set is finite and countable,
its elements can be ranked. Let the tiebreaker function, τ(x), denote the rank
assigned to the deterministic allocation x. Choose the deterministic assignment in
the linear program’s arg max with the lowest τ value. The following claim shows that
this is enough to yield a well-defined mechanism
Claim 6. The arg max of the v-rank-value mechanism is the convex hull of some set
of deterministic assignments.
Proof. This is a linear optimization on a compact set, so the arg max is not empty.
Consider a lottery representation of some assignment in the arg max. By the linearity
of the problem and the convexity of the feasible set, it must be that the deterministic
assignments in the support are also in the arg max; otherwise, the objective would
increase if they were dropped. Hence, any assignment in the arg max is a convex
combination of deterministic assignments that are also in the arg max. Also, a convex
combination of any set of assignments in the arg max must also be in the arg max,
13
since the convex combination will have the same objective value as its components.
So, the tiebreaker procedure will always choose some element of the arg max, and
if the tie-breaker ordering is drawn from some distribution, the tie-breaker procedure
is effectively implementing a lottery representation of some random assignment in
the arg max. Of course how the tiebreaker is chosen can affect the incentives of the
mechanism. In this paper, I will focus on tiebreaker functions that are drawn from
a distribution that does not depend on the submitted preferences of the agents. The
simplest example of this is to choose the tiebreaker uniformly at random from the
set of all strict orderings over the deterministic assignments. This is the same as
finding all deterministic assignments in the arg max and picking one uniformly at
random, which in turn, is the same as implementing the random assignment that is
the centroid of the arg max.
Regardless of the tiebreaker, however, the family of rank-value mechanisms and
the set of rank efficient assignments are very closely related.
Theorem 1. x is a rank efficient assignment if and only if there exists a valuation
v such that x is supported by the v-rank-value mechanism.
The intuition of this result comes from thinking about assignments in rank dis-
tribution space, that is mapping an assignment x to (Nx(1), Nx(2), . . . , Nx(|O|)) ∈R|O|.25 In rank distribution space, the proof of the theorem is similar to the proofs
for the first and second welfare theorems. For instance, the “if” part comes from
rewriting the objective of the v-rank-value mechanism as26
|O|−1∑k=1
Nx(k) (vk − vk+1) + |A| · v|O|
By definition, if x is rank-dominated by x, then x will increase this objective, since
vk > vk+1. The “only if” part comes from an argument that uses the non-standard
polyhedral separating hyperplane theorem of McLennan (2002).27 The details of the
proof, I relegate to the Appendix.
The rank-value mechanism is not strategy-proof; in fact, in Section 8 I show
that no mechanism can be both rank efficient and strategy-proof. I will eventually
25Since the mechanism is individually rational, it will never assign any agents to their (|O|+ 1)th
choice, since ∅ is never ranked below |O|th.26Nx(|O|) = |A| by definition, since all agents are assigned to some object (even if it is ∅).27The non-standard theorem is necessary to ensure that vk is strictly larger than vk+1 for all k.
14
address this shortcoming, but for now I merely state it as fact. Also note that linear
programming mechanisms have been previously observed in the field (Roth 1991) and
are presently used by Teach for America (see Section 4.4 for more).
4.3 Sequential improvements
As I have previously mentioned, ex post efficiency is a Paretian concept, that is, it
only looks for improvements that make all agents weakly better off. Rank efficiency,
on the other hand, is equipped with a method for looking at non-Pareto improvements
and deciding whether the good outweighs the bad. In other words, rank efficiency
can make “tough decisions” while ex post efficiency cannot. Thinking about rank
efficiency in terms of trade cycles can help to make this intuition more precise. For-
mally,
Definition 6. A trade cycle on assignment x is a sequence τ = ((a1, o1) , . . . , (am, om))
such that xakok > 0 for each (ak, ok) in the sequence.
Definition 7. A claim-trade cycle on assignment x is a special trade cycle where
a1 is a fictional agent who is indifferent between all objects, and o1 is an object that
has not been entirely claimed by the agents in A, that is, qo1 >∑
a xao1 .
Remark 4. The fictional agent can be interpreted as holding all object shares that
are not claimed by any of the agents in A.
Definition 8. Implementing trade cycle τ on assignment x yields a new as-
signment, x such that all entries stay the same except for (interpreting o0 ≡ om),
• xakok−1= xakok−1
+ Ξ, for all k ∈ 1, . . . , |τ |
• xakok = xakok − Ξ, for all k ∈ 1, . . . , |τ |
where Ξ ≡ mink∈1,...,|τ |
xakok .
Implementing a claim-trade cycle is done similarly, except only the allocations
of agents a2, . . . , a|τ | are changed, and Ξ ≡ min
min
k∈2,...,|τ |xakok , qo1 −
∑a xao1
.
A trade cycle can be interpreted as a sequence of agents who hold probability
shares of objects. Implementing a trade cycle means that ak gives some of his ok to
ak+1 and receives the same amount of ok−1 from ak−1, and so on.28 A claim-trade cycle
28The one-to-one nature of the trade is why the volume of trade through the cycle in the definition,Ξ, is limited by the smallest holding of an agent in the cycle.
15
is just a trade that involves unclaimed shares. In addition to the characterization of
the previous section, rank efficiency can also be characterized in terms of trade and
claim-trade cycles in which some agents are made worse off but others are made better
off. A valuation vector determines whether the good outweighs the bad. Formally,
Definition 9. A trade cycle or claim-trade cycle τ on assignment x is a v-rank-
improving cycle if
|τ |∑k=1
[vrak (ok−1) − vrak (ok)
]> 0 (where o0 ≡ o|τ |).
Remark 5. Note that the k = 1 term in the sum is zero for a claim-trade cycle, since
the fictional agent is indifferent between all objects.
Theorem 2. An assignment x is rank efficient if and only if there exists a valuation
v such that x admits no v-rank-improving cycles.
The “only if” part is straightforward. Note that claim-trade cycles must be con-
sidered because |A| could be strictly less than∑
o qo. Trade cycles cannot remedy
the bad situation in which a valuable object is unassigned.29 The “if” requires some
subtlety. Intuitively, a change from assignment x to assignment x can be decomposed
into distinct trade cycles and acquisitions of unassigned objects. Now say that x
rank dominates x. Then, for any v, the objective of the v-rank-value mechanism is
bigger for x than for x, which by linearity, means that at least one of these cycles or
acquisitions must improve the objective, which is exactly what the condition in the
definition of a v-rank-improving cycle means.
Hence, the valuation vector v provides the rule that the v-rank-value mechanism
uses to decide if trade cycles that aren’t Pareto improving should be implemented. A
corollary to this method of proof (and the finiteness of the market) is that the policy-
maker can find an element of the arg max of the v-rank-value mechanism through a
sequential improvement process:30
1. Start with a feasible assignment matrix.
2. If possible, find a v-rank improving cycle and implement it.
29Bogomolnaia and Moulin (2001) characterize ordinal efficiency (see Section 6) in terms of tradecycles that improve all agents. They are able to dispense with the fictional agent because theirmodel assumes |A| = |O|. In Proposition 7, I extend their theorem to the setting of this paper.
30At least theoretically. Although this algorithm will eventually terminate, it may take a longtime: Teach for America would spend about two man-weeks to match around 2,000 teachers usinga method like this (they did this 4 times a year). It is unclear if they stopped because they couldn’tfind any more rank-improvement cycles, or if they were merely quitting due to time constraints. Itis true, however, that each step of the process necessarily increases the value of the objective of thev-rank-value mechanism.
16
3. If there was no v-rank improving cycle, then terminate. Otherwise, go to Step 2
with the new assignment.
Proposition 4. If all elements of the initial assignment matrix are rational numbers,
then the sequential improvement process will terminate in a finite number of steps with
a rank efficient assignment that is in the arg max of the v-rank-value mechanism.
Remark 6. All entries of a deterministic assignment matrix are rational.
In other words, starting with a deterministic assignment, a process of sequential im-
plementation of v-rank-improving cycles will converge to a rank efficient assignment.31
While many algorithms for solving the assignment problem are not intuitive enough
for most policy-makers to come up with on their own, the sequential improvement
process is something that many policy-makers consider naturally. This was exactly
what HBS considered doing when it found that too many MBAs were assigned to a
country that they ranked last (see the first paragraph of the Introduction), and it is
exactly how Teach for America conducted their assignment process prior to 2011 (see
the next subsection).
Finally, I will mention that these non-Paretian, “tough decisions” cycles are closely
related to the concept of callousness brought up by Budish and Cantillon (forthcom-
ing) and Budish (2009) in the context of multi-unit assignment. Consider the example
of Figure 1a. Putting weight on the deterministic assignment x can be interpreted
as the mechanism allowing 1 to callously take what he prefers without considering the
greater good. Rank efficiency prevents such callousness, and in fact, implementing
v-rank-improving cycles can be interpreted as correcting callousness ex post.
4.4 Teach for America: before and after
The story of Teach for America’s recent redesign of their assignment system sheds
light on how the characterizations of rank efficiency fit together.32 Teach for Amer-
ica (TFA) is a nationwide non-profit that puts selected college graduates into at-risk
schools to teach. In the 2011-2012 school year, TFA intends to assign around 8,000
teachers nationwide. The application process for potential TFA admits is arduous
31This algorithm is a version of the Klein cycle-canceling algorithm, a primal method used moregenerally to solve minimum cost flow problems (Klein 1967). Although picking improvements ran-domly can have bad time-complexity properties, more sophisticated algorithms (Cunningham and
Marsh 1978) based on this general idea can be O(
(∑
o qo)3)
.32Al Roth and I have been helping Teach for America to redesign their assignment system since
Spring of 2011.
17
and extends over several rounds, including interviews and tests of teaching ability.
Once the original pool of around 50,000 applicants has been reduced to about 20,000,
the applicants submit a final round application, along with rankings over the regions
to which TFA could potentially assign them.33 From these applications, the admis-
sions team decides who should be made an offer without considering their regional
preferences.34
Before the 2010-2011 admissions cycle, TFA used an interesting system to assign
the admits. First, a computer would choose an initial match. How exactly the com-
puter did this is immaterial, but what is important is that the admissions team found
the computer match to have an unacceptably bad rank distribution. To correct this,
two staff members would spend about a week in a conference room looking for trad-
ing cycles that could improve the match. Almost all of these were cycles where some
admits were made worse off, while the rest were made better off. The whole process
closely resembles the sequential improvement characterization in Section E.1.2.35
In the course of the redesign, we observed the old process and tried to recreate it
with an linear program. This was quite successful, which is no surprise in the light
of the previous characterizations. Of course, we worried about strategy-proofness
and strongly cautioned TFA about the potential problems of manipulation, but they
had a strong belief that TFA admits wouldn’t or couldn’t game the system. They
are geographically separated, only rank the regions once, and know little about the
relative popularities of the regions. I formalize a version of this line of reasoning in
Section 8. Also, TFA staff found the rank distributions that resulted from random
serial dictatorship simulations inferior. In Section 7, I show that rank efficient mech-
anisms can yield rank distributions that are markedly better than those yielded by
random serial dictatorship.
Although the linear program was never intended to be what TFA should use,
TFA pressed for it, not because it offered anything new, but because it accomplished
in 30 seconds what had before taken two man-weeks.36 The perceived equivalence
33There were 43 regions in the 2011-2012 assignment cycle.34When Al and I suggested that assignment and admissions might be integrated, the idea was
quickly rejected. Teach for America separates admission from assignment for two reasons. The firstis that they feel they should admit the best candidates, and that it would be unfair to an unpopularregion to lower the bar in finding its teachers. The second is that Teach for America feels that itcan successfully persuade many admits into accepting their assignment, even if it isn’t a top choice.
35TFA does not allow admits to rank ∅, and the number of admits was equal to the number ofpositions. In this environment, all feasible assignments are non-wasteful.
36The old rank-improvement cycles method was mainly implemented by Teach for America’sDirector of Assignment Strategy, Johann von Hoffman. The new linear program is known colloquiallyat TFA as the “auto-Johann”.
18
between the automated and non-automated processes used by TFA turns out to line
up quite well with the theoretical results we have established. So, although TFA has
only recently started using a linear program to run their match, the non-automated
system they used in the past also seems to have been a rank efficient mechanism.
5 Choosing the valuation vector
For rank-value mechanisms to be used practically, it is important to know what the
valuation vector represents. In this section, I will give several interpretations of the
rank-value mechanism that shed some normative light on what the valuation vector
should be.
5.1 Scoring interpretation
Rewriting the objective of the rank-value mechanisms as
∑k
vk ·
[∑a
∑o
1ra(o)=k · xao
]︸ ︷︷ ︸expected number of agents
who get their kth choice
yields an easy interpretation. Score an assignment by giving it v1 points for every
agent (in expectation) who gets his first choice, v2 points for every agent who gets
his second choice, and so on. Then, look for the feasible assignment with the biggest
score. This is an easy to understand explanation for non-economists, and it can have
some justification depending on where v comes from.
For instance, let the objects be job offers. If vk is the policy-maker’s belief about
the probability that an agent will accept the offer to his kth choice job, then the rank-
value mechanism is merely maximizing the expected number of agents who accept
the offer. This is exactly the problem faced by Teach for America, where admits
get notification of admission and assignment simultaneously (applicants submit pref-
erences over the regions when they apply, before they are admitted) and are not
allowed to rank regions as unacceptable.37 Currently, TFA is considering modeling
37TFA does not allow this, as they feel they have some probability of convincing any agent toaccept any assignment. In fact, after notification goes out, TFA regional directors call some admitsdirectly to persuade them to accept.
19
acceptance probabilities for use in their mechanism, which indicates that they take
this interpretation of the rank-value mechanism seriously.
Alternatively, a policy-maker might want to minimize the expected rank in the
assignment market. If he sets vk = |O| − (k − 1), the rank-value mechanism will
accomplish this, since such a valuation makes the objective equal to |O|+ 1−∑
k k ·[∑a
∑o 1ra(o)=k · xao
], where the term in square brackets is the expected number of
agents who get their kth choice under assignment x.
Although these interpretations might seem a bit ad hoc, they can be useful in
appropriate situations. A more general interpretation of the valuation vector is dis-
cussed in the next subsection: v can be thought of as an assumption about the
cardinal utilities of the agents.
5.2 Ex ante welfare interpretation
When evaluating policy, it is often helpful to look at things from behind the veil-of-
ignorance, that is, from the perspective of a fictional agent with no preferences of his
own, who knows that he will randomly become one of the agents in the assignment
market, inheriting her preferences and allocation (Rawls 1972). From this “original
position”, Harsanyi (1975) proposes that a rational agent should act to maximize his
expected utility. Assuming the von Neumann-Morgenstern axioms for the fictional
agent, this gives an expected utility representation over agent-object pairs (Harsanyi
1955, 1986) which is functionally a weighted sum of the expected utilities of the
individual agents, that is, a social welfare function, W =∑
a αa∑
o ua(o).
The ua’s encode how a values gambles over objects; these are objective statements
that can be falsified.38 The α’s, however, encode how the fictional agent will evaluate
gambles concerning outcomes in which he gets the same allocation, but as different
agents. Since the original position is merely a thought experiment about justice, the
α’s encode ethical judgments that cannot be falsified. Generally, economists avoid
statements about interpersonal utility comparison, not because they aren’t theoreti-
cally grounded, but because of an aversion to non-falsifiable ethical judgements. Mar-
ket designers, however, often need to show policy-makers how to input their ethical
judgements into a mechanism.39
38For instance, one could ask agent a the following question: “Option A is a lottery where x% ofthe time you get your 1st choice and (100− x)% of the time you get your 3rd. Option B is gettingyour 2nd choice with certainty. What is the lowest x at which you prefer Option A?”
39A more general version of the rank-value mechanism, which would allow for a policy-maker tomake whatever assumption about interpersonal utility comparison he likes, is discussed in Section 9.1.
20
Considering the objective of the rank-value mechanism as a welfare function, the
fact that all agents’ cardinal utility profiles are derived from the same valuation vector
seems to assume that all agents have precisely the same cardinal utility functions
(modulo differences in ordinal ranking). Fortunately, this is not the case. Let vao
be agent a’s von Neumann-Morgenstern utility value for object o, and let F be a
distribution over these values for all agents and objects. F encodes the planner’s
beliefs about cardinal utilities and interpersonal utility comparison, once he has seen
the submitted ordinal preferences. Given F , to maximize welfare from behind the
veil-of-ignorance is to maximize
ˆ ∑a
∑o
∑k
vak · 1ra(o)=k · xao · dF (v)
Note that it is without loss of generality to index v by agent, a, and rank, k. The
following proposition clarifies what assumptions justify running a rank-value mecha-
nism.
Proposition 5. Assume the planner wants to maximize welfare subject to beliefs
encoded by the distribution F . If, relative to F , the unconditional expectation of vak is
independent of a, then the planner can do so by running the v-rank-value mechanism,
where vk =´vak · dF (v) (for any agent a).40
Proof. Changing the order of summation (and integration) in the welfare, we find
∑a
∑o
∑k
[ˆvak · dF (v)
]· 1ra(o)=k · xao
=∑a
∑o
∑k
vk · 1ra(o)=k · xao =∑a
∑o
vra(k) · xao
Note that the condition of the Proposition can be interpreted either as a direct
assumption, or as an informational limitation, i.e. the planner would have different
beliefs for the agents if he knew more about them, but he doesn’t. Also, note that the
rank-value mechanism weights all agents the same, that is, it places the same social
value on any agent getting a kth choice. This ethical judgment, while reasonable, is
replaceable (see Footnote 39).
40See the Appendix for a generalization of this proposition to different assumptions about howagents’ utilities compare.
21
A1 : a b c
A2 : a b c
B1 : b a c
B2 : b a c
(a) Preferences
a b c
A1 : 5/12 1/12 1/2
A2 : 5/12 1/12 1/2
B1 : 1/12 5/12 1/2
B2 : 1/12 5/12 1/2
(b) Uniform RSD assignment
a b c
A1 : 1/2 0 1/2
A2 : 1/2 0 1/2
B1 : 0 1/2 1/2
B2 : 0 1/2 1/2
(c) Improvement
Figure 2: Example from Che and Kojima (2010)
Of course, different assumptions could also have been made concerning the under-
lying cardinal utilities. If the policy-maker could observe each agent’s “type”, then he
might want to assume different valuation vectors for each type. Maximizing ex ante
welfare would still be solved by a linear program under such assumptions, however,
the resulting assignment would not necessarily be rank efficient. In fact, this devia-
tion from rank efficiency could be desirable under some informational assumptions.
For instance, rank efficiency requires implementation of a bilateral trade that takes
a1 from 1st choice to 2nd and a2 from 3rd to 1st. But if it were known that a1 values
his 1st choice much more than his 2nd and that a2 is close to indifferent between his
top three choices, this trade would be bad for welfare. In this sense, rank efficiency
makes more intuitive sense under symmetric informational assumptions.
6 Relation to ordinal efficiency
Thus far, I have considered rank efficiency as a refinement of ex post efficiency. In this
section, I will introduce another refinement commonly41 discussed in the literature,
ordinal efficiency, and show that rank efficiency is a refinement of it as well. I will
then present two propositions that highlight how ordinal efficiency differs from rank
efficiency.
Consider the example in Figure 2a, adapted from Che and Kojima (2010). There
is one copy each of objects a and b, while there are two copies of object c. Random
serial dictatorship relative to the uniform distribution over all orderings yields the
random assignment in Figure 2b which, by construction, is ex post efficient.42 If
41For instance, see McLennan (2002), Abdulkadiroglu and Sonmez (2003a), Manea (2008), Manea(2009), Che and Kojima (2010), and Kojima and Manea (2010).
42To derive this random assignment matrix, note that only the first two agents in the orderingget a or b; hence, every agent gets c half the time. For an agent a not to get his preferred object,he must be second in the ordering (1/4 of the time) and, conditional on that, the first agent in theordering must take a’s first choice (1/3 of the time). (1/4)× (1/3) = 1/12.
22
each row of the assignment matrix represents a bundle of probability shares, however,
there is a mutually beneficial trade: A1 and A2 can trade their shares in b for the
shares of a held by B1 and B2, yielding the assignment in Figure 2c.43 Bogomolnaia
and Moulin (2001) first noticed this problem and suggested a refinement of ex post
efficiency which remedies it. Define the personal rank distribution of agent a
by Nx,a(k) ≡∑
o∈O 1ra(o)≤k · xao.
Definition 10. A feasible random assignment x ordinally dominates another as-
signment x if for all agents a, xa weakly stochastically dominates xa with respect to
%a, that is, N x,a(k) ≥ Nx,a(k) for all a ∈ A and k (strict for some a and k). An
assignment is called ordinally efficient if there is no other feasible assignment that
ordinally dominates it.
Bogomolnaia and Moulin (2001) showed that ordinal efficiency is a natural interim44
refinement of ex post efficiency, just as I earlier showed that rank efficiency can be
interpreted as an ex ante refinement of ex post efficiency. It is also true that rank
efficiency is a refinement of ordinal efficiency.
Proposition 6. If x is rank efficient, then x is ordinally efficient; however, the
converse need not hold.
Proof. The example from Figure 1a shows that the converse need not hold. For the
forward implication, note that weakly first-order stochastically improving all agents
(strict for one) will necessarily lead to a first-order stochastic improvement in the rank
distribution, since the global rank distribution is a sum over the agents’ individual
rank distributions.
Just as the random serial dictatorships always produce ex post efficient assign-
ments, the simultaneous eating mechanisms always yield ordinally efficient as-
signments (and can generate any ordinally efficient assignment). The formal descrip-
tion of this class of mechanisms is in Section C of the Appendix, but, essentially,
agents simultaneously claim object shares (“eat”) at an agent-specific, predefined
rate from their favorite (remaining) object, changing objects only when their current
43The Budish et al. (2011) extension of the Birkhoff-von Neumann theorem guarantees that thisnew assignment can also be resolved into a lottery over deterministic assignments. In fact, so longas the trades are one-for-one, this is true; the one-for-one requirement ensures that no agent endsup with shares that sum up to strictly more or less than one.
44Interim denotes the information set from which agent types are known, but the realized de-terministic assignment is not. In the context of assignment markets, this is the perspective whereagents’ ordinal preferences are known and the lottery over deterministic assignments has yet to beresolved.
23
favorite is exhausted.45 The probabilistic serial mechanism (PS) is the member
of this class where all agents “eat” at a uniform rate; it is the anonymous simultane-
ous eating mechanisms, just as uniform random serial dictatorship is the anonymous
random serial dictatorship. It is important to note that simultaneous eating mech-
anisms are generically non-strategy-proof and are absent in the field. Since ordinal
efficiency is such an attractive theoretical concept, it was unclear why there were no
ordinally efficient mechanisms in the field.46 The previous proposition, coupled with
the fact that rank efficient mechanisms do exist in the field resolves the puzzle. In-
stead of ordinal efficiency most broadly, institutional evolution seems to have chosen
the rank efficiency refinement. Table 1 summarizes the relationships between ex post
efficiency, ordinal efficiency, and rank efficiency.
Although ordinal and rank efficiency are closely related, two propositions highlight
the differences. First, much like rank efficient assignments can be characterized in
terms of rank-improving cycles, ordinally efficient assignments can be characterized
in terms of improvement cycles, that is, trade cycles in which ok−1 %ak ok for all
k (strict for some k).
Proposition 7 (Bogomolnaia and Moulin 2001). An assignment x is ordinally effi-
cient if and only if x is non-wasteful and admits no improvement cycles.
The proof is a straightforward extension of Bogomolnaia and Moulin (2001) and is left
to the Appendix. Comparing this characterization to that of Section 4.3 highlights
the fact that rank efficiency is a non-Paretian concept
Ex post Interim Ex ante
Efficiencyconcept
Ex post efficiency⇐;
Ordinal efficiency⇐;
Rank efficiency
MechanismsRandom serialdictatorships
Simultaneouseating mechanisms/probabilistic serial
Rank-valuemechanisms
Table 1: Relationships between efficiency concepts
45In Section C of the Appendix, the assignment resulting from running PS on the example fromFigure 1a is derived as an example. It is also rank-dominated by x©.
46Serial dictatorships (one fixed dictatorship ordering) are ordinally efficient, but vacuously so, asall ex post deterministic assignments are also ordinally efficient (see Section E.1.2). More preciselythen, the puzzle is why there are no ordinally efficient mechanisms that satisfy equal treatment ofequals, that is, that give the same allocation to all agents who submit the same preference.
24
Second, conditional on an assignment being deterministic, ordinal efficiency is
equivalent to ex post efficiency, while rank efficiency remains a refinement. Formally,
Proposition 8. Let x be a deterministic assignment. Then,
(i) x is ordinally efficient if and only if it is ex post efficient.
(ii) x is rank efficient implies it is ex post efficient; however the converse need not
hold.
Motivating rank efficiency as ex ante efficiency, modulo a few assumptions, is theoret-
ically pleasing, but policy-makers tend to think in terms of deterministic assignments.
If this is true, then the preceding distinction might shed some light on why simultane-
ous eating mechanisms aren’t seen in the field. Rank efficiency is a strict refinement
even when only deterministic assignments are being compared.
7 The potential gains of the rank-value mechanism
As mentioned previously, both the ordinally efficient and rank efficient mechanisms
are generically non-strategy-proof. This could be a major problem, as agent manip-
ulation of reported preferences could yield assignments that look good relative to
the submitted preferences, but are quite bad relative to the true preferences. In this
section, I consider a simple empirical exercise that sheds light on this risk. Running
uniform random serial dictatorship (the baseline strategy-proof mechanism) gives an
idea of how efficient a match can be without sacrificing strategy-proofness. The pref-
erences submitted to a strategy-proof mechanisms should be truthful, so to get an
idea of the potential for gains, I can take those preferences and use them to run a non-
strategy-proof alternative, such as the probabilistic serial mechanism or a rank-value
mechanism. The efficiency of these counterfactuals is a sort of best-case analysis:
if there is not significant improvement even without manipulation, then sacrificing
strategy-proofness is likely not worth considering.
7.1 Assigning students to overseas programs at Harvard Busi-
ness School
At Harvard Business School (HBS), first year MBAs must participate in a global
immersion program.47 They are assigned to a foreign company and remotely work
47At HBS, it is known as the FIELD 2 program.
25
on a project with that company during the first semester. The program culminates
in a two-week trip over the winter break in which the MBA will present her work in
person and be given the opportunity to make foreign business contacts.
In 2011, at the beginning of their first semester at HBS, 900 MBAs were asked
to rank the 11 different countries to which they could be assigned.48 The mechanism
we49 used to match the students was strategy-proof, so the preferences submitted to
it can be taken as truthful. Note that we allowed the MBAs to express indifferences;
the strategy-proof adaptation of random serial dictatorship that we used is briefly
described in the Appendix. In the interests of simplicity, however, I randomly broke
student indifferences and considered only strict preference assignment mechanisms.
All results are qualitatively the same regardless of the method used to break student
indifferences, and regardless of whether indifferences are broken. Results from these
alternate specifications are included in the Appendix.
7.2 The data
The data used for the analysis in this section is borrowed from Featherstone and
Roth (2011). Figure 3 shows the rank distributions yielded by uniform random serial
dictatorship, probabilistic serial (note that these first two overlap on the graph), and
the v-rank-value mechanism, where the valuation vector50 is
v = (100, 80, 50, 35, 15, 10, 5, 3, 2, 1, 0.5)
The first thing to notice is that the rank distribution from probabilistic serial is
essentially identical to that of uniform random serial dictatorship. The underlying
assignments are also very similar, which means that, in the HBS context, there is
little reason to favor the more complicated probabilistic serial mechanism over the
simpler uniform random serial dictatorship. Che and Kojima (2010) show that, the-
oretically, the random assignment generated by the probabilistic serial mechanism
asymptotically converges to the random assignment generated by uniform random
serial dictatorship in the large market limit. Often with asymptotics, however, it is
hard to know how large is large enough. In the case of HBS, 900 students and 11
48Students only rank countries; once they are assigned to a country, the company they get isadministratively assigned without their input.
49I designed the HBS match jointly with Al Roth.50Other valuation vectors yield similar results; the main purpose I document the specific valuation
used is to assure the reader that the analysis in this section doesn’t depend on a particularly extremechoice of v.
26
countries seems sufficient.
Perhaps more striking is how well the v-rank-value mechanism performs. The
number of students who get their first or second choice is increased by more than
15% when we move from the uniform random serial dictatorship to the rank-value
mechanism. This corresponds to about 120 students. So, the gains from moving
to a rank-value mechanism might indeed be large enough to justify backing away
from strategy-proofness. However, these figures come from a best-case analysis. It
could be that manipulation completely undermines the gains, or even makes the rank-
value mechanism perform worse than random serial dictatorship.51 The analysis of
this section simply makes the case that backing away from strategy-proofness for the
gains given by probabilistic serial does not make sense for the HBS match, but that
doing so for the gains given by the rank-value mechanism might well be worthwhile.
7.3 Strategy-proofness versus improved efficiency
The previous exercise illustrates that the concept of rank efficiency and its asso-
ciated rank-value mechanisms fit well with previous literature that has considered
the costs of strategy-proofness in stable matching (Erdil and Ergin 2008, Abdulka-
diroglu, Pathak and Roth 2009, Azevedo and Leshno 2010). Papers on the costs of
strategy-proofness in non-stable assignment have mostly focused on how the Boston
mechanism (Abdulkadiroglu and Sonmez 2003b) can outperform random serial dicta-
torship. Featherstone and Niederle (2011) show this in the context of a truth-telling
equilibrium, while Abdulkadiroglu, Che and Yasuda (2011) focus on a non-truth-
telling equilibrium. Both papers rely on stylized assumptions about how preferences
are distributed. The present paper differs in two ways. First, it offers a new approach
to get at cardinal utility in the context of an ordinal mechanism: make reasonable
assumptions and maximize welfare accordingly. Second, in past literature, the leading
contender with random serial dictatorship has been the class of Boston mechanisms.
This paper offers a new mechanism that is based on the well-established empirical
fact that policy-makers care about rank distributions.52,53 As I have begun to show in
51Note that the work we do in Section 8 indicates that in at least some environments, we expecttruth-telling.
52Note that the Boston mechanism is not entirely ad hoc: it has been axiomatized by Kojimaand Unver (2011).
53Also, notice that the Boston mechanism is not a v-rank-value mechanism for any v. To see this,consider the example of Figure 1a. The Boston mechanism would give object a to 1 half the time, andto 2 the other half of the time. But, Claim 2 shows that this is not a rank efficient assignment, andTheorem 1 shows that rank-value mechanisms must yield rank efficient assignments. The Bostonmechanism is not forward-looking (i.e. it is a greedy algorithm), which is a requirement for any
27
60%
65%
70%
75%
80%
85%
90%
95%
100%
1 2 3 4 5 6 7 8 9 10 11
Ran
ked
kth
or
bet
ter
k
U-RSD
PS
RVM
Figure 3: Rank distributions for our mechanisms(U-RSD is Uniform Random Serial Dictatorship, PS is Probabilistic Serial, and RVM is Rank-Value
Mechanism. Note that U-RSD and PS overlap.)
28
this section, the rank-value mechanisms might be an important part of the discussion
about the costs of strategy-proofness in ordinal assignment markets.
8 Incentives for truth-telling
I begin this section by showing that efficiency and strategy-proofness are theoretically
incompatible. There are two ways that rank-value mechanisms could still work in light
of this. First, there could be environments in which rank-value mechanisms admit
manipulating equilibria in which the efficiency gains don’t disappear, much like in
Abdulkadiroglu, Che and Yasuda (2011). Second, there could be environments in
which truth-telling can be supported in equilibrium. Although, the first possibility is
interesting, in this paper, I will focus on the second.
8.1 An impossibility result
A mechanism is called strategy-proof if the allocation it gives to any agent when
she truthfully reveals her ordinal preference stochastically dominates the allocation it
would give her if she revealed anything else. A mechanism is called weakly strategy-
proof if the allocation it gives to an agent when she deviates from truth-telling never
strictly stochastically dominates what it gives her when she truthfully reveals. An-
other way to phrase the difference between these two concepts is in terms of the car-
dinal preferences that rationalize the true ordinal preferences. Strategy-proof means
that truth-telling is a dominant strategy regardless of the rationalizing cardinal util-
ities. Weakly strategy-proof means that for any beliefs and any potential lie, there
exist rationalizing cardinal utilities that make truth-telling a better-response.
Theorem 3. No rank efficient mechanism is strategy-proof. In fact, no rank efficient
mechanism is even weakly strategy-proof.
Proof. Consider a four agent, four object example.
1 : b e
2 : b c e
3 : c d e
4 : d b
algorithmic solution to the defining linear program of a rank-value mechanism.
29
The unique54 rank efficient allocation is (1, e) , (2, b) , (3, c) , (4, d). So under any
rank efficient mechanism, 1 must be assigned to e. Now, consider what happens if
1 deviates from truth-telling and submits b c d e instead. Now, the unique
rank efficient allocation is (1, b) , (2, e) , (3, c) , (4, d). Hence, under any rank efficient
mechanism, 1 gets b with the deviation, which first-order stochastically dominates the
certain e he would get with the truth.
Intuitively, someone has to get stuck with e. Under the true preferences, the
mechanism pays the smallest price for assigning e to Agent 1. When he submits
b c d e instead, he is exaggerating about how bad e is for him, making it
better for the mechanism to give it to Agent 2 instead. Still, this example required
quite a bit of specific knowledge. Next, I will show that truth-telling is a natural
response in some environments where agents have much less information.
8.2 A possibility result
Following Roth and Rothblum (1999), for some preference profile %, define %o↔o′ to
be the same preference profile, except all agents have switched the objects o and o′
in their ordinal rankings. Define qo↔o′
to switch the capacities of o and o′, and let
xo↔o′
denote the (possibly infeasible) assignment where everyone assigned to o in x
is reassigned to o′ and vice-versa. Finally, define the action of the o ↔ o′ operator
on the tiebreaker by τ(x)o↔o′ ≡ τ
(xo↔o
′). Note that xo↔o
′will be feasible if the
capacities are switched to qo↔o′. An agent a’s beliefs are then summarized by a
vector of random variables that range over potential ordinal preferences of the other
agents (%−a%−a%−a), capacities of the objects (qqq), and tie-breaker numberings, (τττ). We treat
%a, A, and O as known.
Definition 11. An agent’s beliefs are o, o′-symmetric if the distributions of
(%−a%−a%−a, qqq, τττ) and (%−a%−a%−a, qqq, τττ)o↔o′
coincide. If the beliefs are o, o′-symmetric for all
o, o′ ∈ O \ ∅, then we simply call the beliefs symmetric.
One interpretation of this condition is that it represents a very symmetric envi-
ronment. The interpretation I favor, however, is that it represents an environment in
which agents have very little specific information, that is, an environment in which,
to the agents, nothing really distinguishes one object from the other. Even when this
is not globally true, it is often the case that preferences are tiered and that, within a
54For an example of how to show that an assignment is the unique rank efficient assignment, seeClaim 2 above.
30
tier, beliefs are close to symmetric. Note that this condition is well defined even for
preferences that have indifferences. Under this informational assumption, switching
the order of two objects in the submitted preference is not profitable. Formally,
Proposition 9. Under a rank-value mechanism, if agent a’s beliefs are o, o′-symmetric,
and o′ a o, then for a, the allocation resulting from any submitted preference that
declares o a o′ is weakly stochastically dominated by the allocation resulting from a
submitted preference that does not. If the beliefs are symmetric, then this holds for
all o, o′ ∈ O \ ∅.
The proof for the theorem leverages the symmetry assumption to show that for
every state of the world in which the lie is profitable, there is at least one equally
likely state of the world in which it is not. This result is stated in a way that allows
for indifference, but from this point on in the main text, for expositional ease, I will
only consider assignment markets where all preferences are strict. For the interested
reader, the more general results for indifferences are in the Appendix. An immediate
corollary of Proposition 9 is that if an agent has strict preferences and must rank all
objects, then he does best to truth-tell if his beliefs are symmetric. Formally,
Theorem 4. Under a rank-value mechanism, if an agent’s beliefs are symmetric, his
preferences are strict, and he is required to rank all objects, then the allocation he
receives under truth-telling weakly stochastically dominates the allocation he receives
under any other strategy.
Corollary to Theorem 4. Under the conditions of Theorem 4, all agents truth-
telling is an equilibrium.
Although the requirement to rank all objects may seem unrealistic, it is actually
quite prevalent in the field. Consider public school assignment. Students are not
required to rank all schools, but if they cannot be assigned to a school they ranked,
they are generally given an administrative assignment.55 In this sort of situation,
failing to rank all schools is equivalent to the student saying to the school district,
“beyond what I ranked, you can fill out the rest of my rank-order list for me.” As
an aside, Teach for America also does not allow agents to rank the null object: any
admit will get some assignment with their offer of admission.
55I know that this is the policy in the San Francisco Unified School District because I and co-authors (Atila Abdulkadiroglu, Muriel Niederle, Parag Pathak, and Al Roth) spent a year redesigningthe assignment system; however, it is hard to imagine a school district that doesn’t work in thisway, as all students legally must be able to claim some seat in a public school, even after the matchhas run.
31
8.3 Characterizing exhaustive manipulation classes
Nevertheless, one might feel that forcing agents to rank all objects is too onerous
an assumption. If I relax this requirement, I can still tightly characterize classes
of manipulations with which an agent can always best respond. The surprise of
this exercise is that it hinges on an otherwise unremarkable detail of the design.
Implementing mechanisms in the field often focuses attention on things that might
otherwise be assumed away, and the results in this section are a result of carefully
considering what a rank-value mechanism should look like in the context of the HBS
global immersion match discussed in Section 7. How the null object is treated in the
rank scheme turns out to be of great import.
To understand manipulation when agents are not required to rank all objects, I
must first resolve the ambiguity concerning rank functions mentioned near the end of
Section 2. First, define a rank scheme to be upward-looking if ra(o) is independent
of the preferences among objects to which o is strictly preferred . Now, consider how
to deal with the outcome of being unmatched in the case of an agent whose true
preferences are a b ∅. One natural alternative is to treat ∅ like any other object,
that is r(a) = 1, r(b) = 2, and r(∅) = 3. Often, however, policy-makers think of
being unmatched as a distinct worst element in the overall rank distribution;56 that
is, r(∅) = |O|, regardless of how many objects the agent declared acceptable. To help
in thinking about this, consider inserting a new indifference class, I, which contains
only o, below all acceptable objects in %a (but ahead of ∅). Call this new preference
%+Ia . If r%+I
a(o) = r%a(∅), then call the ranking scheme unmatched-neutral, as ∅ is
treated just as any other object. If r%+Ia
(∅) = r%a(∅), then call the ranking scheme
unmatched-distinct. In this language, the example where ∅ is treated just like
any other object is unmatched-neutral, and the example where ∅ is always given the
lowest rank possible is unmatched-distinct.
Now, I define two important classes of manipulations. Call a submitted preference
%′a a truncation of the true preference %a if there is some object o %a ∅ such that
%o↔∅a is equivalent57 to %′a above ∅. Similarly, call a submitted preference %′a an
extension of the true preference %a if there is some object ∅ %a o such that %o↔∅a is
equivalent to %′a. Under these definitions, truth-telling is both a truncation and an
extension (when o = ∅).
Theorem 5. If an agent’s beliefs are symmetric, his preferences are strict, and the
56See Footnote 17.57In the rank-value mechanism, how things are listed below ∅ does not matter because ∅ is never
scarce.
32
rank scheme is upward-looking and...
• ...unmatched-distinct, then his allocation from any other strategy is weakly stochas-
tically dominated by his allocation from playing some truncation.
• ...unmatched-neutral, then his allocation from any other strategy is weakly stochas-
tically dominated by his allocation from playing some extension.
Note that I can’t leverage both parts of the theorem simultaneously to find a condition
for truth-telling.
Claim 7. A ranking scheme cannot be both unmatched-neutral and unmatched-
distinct.
Proof. If it is unmatched-neutral, then r%+Ia
(o) = r%a(∅), which means that r%+Ia
(∅) >r%a(∅), contradicting the assumption that the ranking scheme was unmatched-distinct.
Truncations are a familiar class of strategies,58 and the intuition for why they can
be profitable is simple. When the ranking scheme is unmatched-distinct, a truncation
can be interpreted as an agent threatening the linear program: “give me something
I like, or pay the price of giving me ∅”. When the ranking scheme is unmatched-
neutral, this threat no longer works, as truncating actually relaxes the optimization.
So why extensions? Intuitively, consider an agent whose preference is a b ∅ in a
market with 100 other agents whose preferences are a b c d e f ∅. If
the agent doesn’t extend, the linear program will assign him to ∅, since doing so is
worth v3 − v7 more than the alternative of putting some other agent in ∅.So even in low information environments, agents can gain, but it is notable that
they gain by using a class of strategies very similar to those that can theoretically
be used to game two-sided deferred acceptance in a low information environment. In
the lab, Featherstone and Mayefsky (2011) show that subjects can fail to truncate
under a deferred acceptance mechanism, leaving a significant amount of money on the
table by doing so. It would be interesting to see if a similar result held for rank-value
mechanisms,59 and in fact, the experimental and computational work of Unver (2001)
58The paper in which our informational assumption was first used, Roth and Rothblum (1999),found that, in symmetric environments, agents on the proposed-to side of a deferred acceptancemechanism can always do just as well with a truncation.
59The theorems just proved can help in the design of experiments that deal with the large strat-egy space of ordinal mechanisms, by allowing the experimenter to focus only on truncations orextensions. For a taste of how this might work, see Featherstone and Mayefsky (2011), which usesthe “truncations are exhaustive” result of Roth and Rothblum (1999) to yield a more analyticallytractable design.
33
and Unver (2005) indicate that this is a real possibility.
9 Connection to competitive equilibrium mecha-
nisms
In order to talk about the fairness properties of rank-value mechanisms, I must first
generalize rank efficiency by allowing agents to be weighted differently in the rank
distribution. In doing so, I will show that some ordinally efficient assignments are
not rank efficient relative to any weights. This generalization is also of independent
interest because it strongly relates to an ordinal adaptation of the much-discussed
pseudomarket mechanism of Hylland and Zeckhauser (1979), in which agents are
given a budget of artificial money and a competitive equilibrium is computed.
9.1 Generalizing rank efficiency
Until now, all agents have been equally weighted in the rank distribution, which
is equivalent to saying that the utility an agent gets from a kth choice is directly
comparable to the utility any other agent gets from a kth choice.60 I will weaken
this assumption by allowing for a general weighting scheme in the rank distribution.
Define the rank distribution of assignment x with respect to weights ααα (or
the α-rank distribution for short) to be
Nxα(k) ≡
∑a∈A
∑o∈O
αa · 1ra(o)≤k · xao
Nxα(k) is the weighted expected number of agents who get their kth choice or better
under assignment x.
Definition 12. A feasible random assignment x is ααα-rank dominated by another
feasible assignment x if the α-rank distribution of x first-order stochastically dom-
inates that of x, that is, N xα(k) ≥ Nx
α(k) for all k (strict for some k). A random
assignment is called ααα-rank efficient if it is not α-rank-dominated by any other
feasible assignment.
The mechanisms that correspond to α-rank efficiency are a similarly modified
version of the v-rank-value mechanisms.
60The interpersonal utility comparison is actually more subtle than this; see Section 5.2 for adiscussion.
34
Definition 13. The rank-value mechanism with respect to weights α and
valuation v (or the (α,v)-rank-value mechanism for short) maps agent rank
orderings to a maximizer of the following linear program.
maxx
∑a∈A
∑o∈O
αa · vra(o) · xao
s.t.∑a
xao ≤ qo, ∀o ∈ O∑o
xao ≤ 1 , ∀a ∈ A
xao ≥ 0 ,∀o
∀a
∈ O
∈ A
We say that the assignments in the arg max are supported by the (α,v)-rank-
value mechanism.
Everything proven thus far can be easily adapted to the generalization; proofs in the
appendix are done with respect to a non-specific α.
9.2 Supportable rank efficiency
Of course, one might not want to make any assumption about how agents’ utilities
compare. The following concept makes this idea precise.
Definition 14. An assignment x is called supportably rank efficient if there exists
an α such that x is α-rank efficient.
Supportable rank efficiency implies ordinal efficiency;61 however, it is perhaps sur-
prising that ordinally efficient assignments can fail to be supportably rank efficient.
Theorem 6. Supportable rank efficiency implies ordinal efficiency. The converse
need not hold.
The intuition behind the negative statement is rooted in appropriate generalization
of the “tough decisions” trade cycles of Section E.1.2. For an assignment to be
supportably rank efficient, there must be one (α, v) pair such that all trade cycles
fail to be (α, v)-rank improving.62 For every trade cycle then, there is a (possibly
61Supportable rank efficiency implies α-rank efficiency (for some α) which, by the easy general-ization of Proposition 6, implies ordinal efficiency.
62In Definition 9, replace∑k
[vrak
(ok−1) − vrak(ok)
]> 0 with
∑k
αak·[vrak
(ok−1) − vrak(ok)
]> 0.
35
empty) region of (α, v) space such that it is not (α, v)-rank improving. This region
can be characterized by a simple inequality. The proof of the theorem starts with an
ordinally efficient assignment and then defines trade cycles until the corresponding set
of inequalities has no solution. There is nothing special about the counterexample;
as an assignment grows large, it has many trade cycles, so unless special provision
is made to keep them all from being (α, v)-rank improving relative to some common
(α, v), it is no surprise that the system of inequalities cannot be satisfied. The proof
suggests a natural conjecture (which I do not prove in this paper): as the market
grows, the size of the set of supportably rank efficient assignments becomes small
when compared to the size of the set of ordinally efficient assignments. Also, note
that there is no reason that probabilistic serial assignments are special; in fact, the
proof to Theorem 6 uses a probabilistic serial assignment to get the ordinally efficient
assignment that forms the counterexample.
Corollary to the proof. Probabilistic serial does not always yield supportably rank
efficient assignments.
This result might seem especially surprising in the light of McLennan (2002) and
Manea (2008), who both find (using non-constructive and constructive methods, re-
spectively) that any ordinally efficient assignment is ex ante efficient relative to some
set of cardinal preferences that rationalize the ordinal preferences. The difference here
is that, as shown in Section 5.2, the concept of rank-efficiency imposes the added con-
straint that all agents must share the same cardinal utility (in expectation) across
ranks.
9.3 Competitive equilibrium mechanisms
An assignment is supportably rank efficient if and only if it can be supported by the
(α, v)-rank-value mechanism for some (α, v). This is a straightforward corollary of
the Generalization of Theorem 1 proved in Section E.1.1 of the Appendix. It is also
true, however, that an assignment is supportably rank efficient if and only if it can
be generated by a certain adaptation of the pseudomarket mechanisms of Hylland
and Zeckhauser (1979). The idea of these mechanisms is to 1) use a valuation vector
to convert agents’ ordinal preferences to cardinal preferences, 2) give each agent a
budget of artificial money, and 3) calculate a competitive equilibrium in the market
where goods are probability shares in the objects. Formally,
Definition 15. The prices and random assignment (x, p), form a budget equilib-
36
rium with respect to valuation vvv and budgets B if, for all a,
xa ∈ arg maxxa≥0
∑o
vra(o) · xao
s.t.∑o
po · xao ≤ Ba∑o
xao ≤ 1
and
xao ∈ arg maxxao≥0
∑a
∑o
po · xao
s.t.∑a
xao ≤ qo ,∀o
A (B,v)-budget mechanism maps ordinal preferences to an assignment that is
part of a (B, v)-budget equilibrium, and the assignment is said to be supported by
the (B, v)-budget mechanism.
The first optimization can be thought of as the agent’s problem, and the second as
the producer’s problem. Note that calculating a budget equilibrium is more difficult
than solving the rank-value linear program.63
Remark 7. The producer’s problem is equivalent to∑a
xao ≤ qo and po > 0⇒∑a
xao =
qo, for all o ∈ O.
The resemblance64 of budget equilibria to the more familiar competitive equilib-
rium suggests that there should also be results resembling the first and second welfare
theorems. Supportable rank efficiency is the efficiency concept needed to complete
the analogy. Formally,
Theorem 7. x is supportably rank efficient if and only if there exist budgets and a
valuation, (B, v), such that x is supported by the (B, v)-budget mechanism.
The intuition for this proof comes from considering the linear programming du-
als of the optimizations involved in the (α, v)-rank-value mechanism and the (B, v)-
budget mechanism. If x is a supportably rank efficient assignment, then it must be
63Budget equilibria can be approximately solved using a method laid out in the appendix ofHylland and Zeckhauser (1979), which in turn grapples with the known-to-be-difficult problem ofapproximating Kakutani fixed points (Scarf 1967, Scarf and Hansen 1973, Chen and Deng 2008).Polynomial time methods are known for markets in which agents are not limited by the
∑o xao ≤ 1
constraint (Nisan 2007), but it is unclear whether such results can be generalized to budget equilibria.64Note that the (α, v)-rank-value mechanism can be decentralized to look like a competitive
equilibrium as well. This decentralization, which I call an (α, v)(α, v)(α, v)-discount equilibrium is definedand characterized in Section E.4 of the Appendix.
37
in the arg max of the (α, v)-rank-value mechanism for some (α, v). Keeping the same
valuation v for the budget equilibrium, the shadow prices associated with the object
quota constraints will decentralize x, so long as each agent a can afford his allocation,
xa, which can be guaranteed by giving him a budget Ba =∑
o po ·xao. For the reverse
implication, start with an assignment x that is a (B, v)-budget equilibrium, keep the
same valuation, and set αa equal to the inverse of the budget shadow price in agent
a’s optimization.65
An important corollary to the proof of this proposition is that, for a given val-
uation v and set of preferences %, there is a natural mapping between weights α
and budgets B. Although not immediately obvious, the proof also shows that as-
signments in the arg max of the procedurally fair (all agents are weighted the same)
v-rank-value mechanism are supported by a budget equilibrium that is not procedu-
rally fair (budgets are not guaranteed to be equal). Similarly, budget equilibrium from
equal budgets gives assignments that are supported by an (α, v)-rank-value mecha-
nism in which the weights are not guaranteed to be equal. In this sense, procedural
fairness is fundamentally different between rank-value mechanisms and budget mech-
anisms. In the next section, I will show how this tension sheds light on the theoretical
incompatibility of rank efficiency and envy-freeness.
10 Justice and rank-efficiency
Looking at the example from Figure 1a, one might object that Agent 1 has been
unfairly singled out. Envy-freeness is the idea that makes this intuition precise. Let an
assignment be strongly envy-free if xa weakly first-order stochastically dominates
xa′ relative to %a, for all a, a′ ∈ A. Let an assignment be weakly envy-free if
there is no a′ ∈ A such that xa′ strongly first-order stochastically dominates xa
relative to %a, for all a ∈ A. Finally, let an assignment be envy-free relative to
some cardinal preferences if each agent weakly prefers his own allocation to all other
agents’ allocations.
Theorem 8. No mechanism is rank efficient and even weakly envy-free
Proof. Consider the example from Figure 1a. x© is the unique rank efficient as-
signment, so any rank efficient mechanism must choose it. But 1 would prefer 2’s
65Extra care must be taken when the shadow price is zero. This case is worked out in more detailin the Appendix.
38
allocation, regardless of the cardinal preferences that rationalize his ordinal prefer-
ence.
This theorem implicitly deals with two philosophical concepts of justice. Dworkin
argues for the dual requirements of ex post efficiency and envy-freeness, which he calls
“equality of resources” (Dworkin 1981).66 Harsanyi, in contrast, argues for policy to
be chosen as if by an expected utility maximizer in the Rawlsian67 original posi-
tion68, a concept which I will call “Harsanyian justice” (Harsanyi 1975). In terms
of these concepts, Theorem 8 says that in assignment markets, equality of resources
and Harsanyian justice cannot be jointly guaranteed. This is no surprise in view of
the proof of Theorem 7. An assignment implemented by a budget equilibrium from
equal budgets is, in some sense, envy-free, since all agents have the same opportu-
nity set. But this same assignment might also be implementable by a rank-value
mechanism with unequal weights, contradicting Harsanyian justice. To move forward
from Theorem 8 then, one must make a choice. As discussed in Section 5.2, the
rank-value mechanism with equal weights guarantees Harsanyian justice (modulo a
few assumptions). But how does one guarantee equality of resources?
Proposition 10. If there exist prices p, a valuation v, and budgets B, such that
Ba = Ba′ for all a, a′ ∈ A and (x, p) is a (B, v)-budget equilibrium, then x is weakly
envy-free and supportably rank efficient. Further, x is envy-free relative to the cardinal
preferences encoded by v.
Proof. Theorem 7 established that budget equilibria are supportably rank efficient.
Now, assume that xa′ strongly first-order stochastically dominates xa relative to %a.
Then, for any v, agent a’s objective in the definition of the (B, v)-budget equilibrium
must be higher at xa′ than at xa. But a could have afforded xa′ by the equal budgets
assumption. This contradicts optimality. Envy-freeness relative to v follows by similar
logic.
So, budget equilibrium from equal budgets yields weak envy-freeness and sup-
portable rank efficiency (which implies ex post efficiency). Although this provides a
66In fact, Dworkin (1981) literally suggests a cardinal version of this paper’s budget equilibriumfrom equal budgets as an embodiment of the idea.
67Although Harsanyi’s conception of justice is based on Rawls’ veil-of-ignorance, the DifferencePrinciple (i.e. insistence on a maximin objective from the original position) places Rawls’ conceptionmore in line with Dworkin’s (Rawls 1972).
68That is, from the perspective of a fictional agent who knows that they will become one ofthe agents in the market, uniformly and at random, and will inherit that agent’s preferences andallocation.
39
partial workaround to the impasse of Theorem 8, weak envy-freeness is not a strong
condition: it only guarantees that there exist rationalizing cardinal utilities that make
the assignment envy-free. Still, the exercise is very similar to how the v-rank-value
mechanism can be justified as maximizing social welfare: in the absence of cardinal
information, make assumptions and push forward. Under such assumptions, budget
equilibrium from equal budgets yields an envy-free assignment. Finally, note that the
probabilistic serial mechanism of Bogomolnaia and Moulin (2001) is strongly envy
free, but as shown in Section 9.2, it may not be ex ante efficient in a world where
beliefs about agents’ cardinal preferences meet the conditions of Proposition 5.69
11 Conclusion
Rank efficiency is a natural concept that is used in the field, and under truth-telling,
rank efficient mechanisms can yield significant welfare gains. Unfortunately, rank
efficiency is also theoretically incompatible with strategy-proofness. This does not
mean, however, that the market designer should freely dismiss it. In the words of
Day and Milgrom (2008),
“...it is customary in mechanism design theory to impose incentive con-
straints first, investigating other aspects of performance only later. It
is, of course, equally valid to begin with other constraints and such an
approach can be useful. To the extent that optimization is only an ap-
proximation to the correct behavioral theory for [agents], it is interesting
to investigate how closely incentive constraints can be approximated when
other constraints are imposed first.”
The situation in this paper is not unique: there is a growing body of literature that
points to the fact that the costs of strategy-proofness can be quite high. So, when
then should the market designer pay this cost? In this paper, I show that truth-telling
is an equilibrium of rank-value mechanism in a stylized, low-information environment.
Such a theorem highlights the important intuition that rank-value mechanisms are
more likely to work when agents know little about the popularity of the objects.
Even so, it seems likely that there exist environments in which the performance
of rank efficient mechanisms will be unacceptably poor. Unfortunately, any such
69Note that the wedge between fairness and efficiency exists in the cardinal setting as well.Competitive equilibrium from equal incomes (Varian 1974, Hylland and Zeckhauser 1979) givesPareto efficiency, but Negishi’s theorem (Negishi 1960) does not guarantee that the correspondingweights in the planner’s problem will be equal.
40
characterization is again likely to be limited to a stylized model, and characterizations
that truly generalize across environments seem intractable. In addition, it is certainly
plausible that in more complicated situations, equilibrium predictions will be not
be born out in the field (see Featherstone and Niederle (2011) and Featherstone
and Mayefsky (2011)). Thus, to understand when the costs of strategy-proofness
are too dear, theoretical results will need to be complemented with more empirical
approaches, such as experiments and learning models.
Regardless of such evidence however, it may be that in the long run, in any
environment in which a mechanism fails to admit a truth-telling equilibrium, agents
will eventually converge to a manipulating equilibrium. Even if this is true, it might
still be worthwhile to consider how much the short run can be extended. Consider the
HBS match discussed in Section 7. In the first year of the match, had we run a rank
efficient mechanism, it is quite plausible that agents would have truthfully revealed
their preferences. We didn’t do this, however, because we were worried about the
MBAs learning to manipulate over time. Taking extra efficiency in the first year,
at the expense of subsequent years, seemed like too much of a “smash-and-grab”
strategy. But what if the learning process were slower, such that the efficiency gains
could be reaped for the first five years? In such a setting, the “smash-and-grab”
strategy starts to sound like a good idea; in fact, one could even imagine running
a rank efficient mechanism for four years and then switching to a strategy-proof
mechanism as learning began to cause problems. Thinking about ways to slow the
speed with which agents learn to manipulate a non-strategy-proof mechanism seems
like an interesting, and thus far, unpursued line of research. When the efficiency cost
of strategy-proofness is high, as I have shown can be the case, this line of inquiry
could prove quite fruitful.
While there are many papers on the costs of strategy-proofness in matching mech-
anisms, the costs of Pareto-agnosticism have not been as carefully considered. In this
paper, I show that if the policy-maker truly considers all kth choice assignments as
interchangeable, then by allowing him to express this, the rank-value mechanism can
yield big welfare gains. For more complicated welfare considerations, the (α, v)-rank
efficient mechanisms or the generalizations of rank-value mechanisms mentioned at
the end of Section 5.2 could help. The lessons from the mechanisms in this paper are
then twofold. First, although economists are not in a position to make ethical judge-
ments about interpersonal utility comparison, policy-makers are. It makes sense then
for market designers to use mechanisms that have a place for policy-maker’s judge-
ments to be inputted, like the α in the (α, v)-rank-value mechanisms. Second, armed
41
with the power to make such judgements, market designers can significantly improve
welfare.
The fact that there are rank efficient mechanisms in the field could mean that
they are successful in some situations. Alternatively though, it might be that well-
intentioned policy-makers, ignorant of incentive concerns, are implementing bad poli-
cies. Market designers need to know whether they should be correcting the impulse
to depart from strategy-proofness or sometimes suggesting that it is a small price to
pay for potentially big welfare gains. Which advice is good very likely depends on the
environment in which the mechanism is meant to serve and on the dynamics of the so-
cial learning process. Understanding when (if ever) market designers can comfortably
suggest a rank efficient mechanism seems like an important open question.
42
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Appendix
A Ex post dominance expressed as state-by-state
dominance
Consider a discrete probability space, (Ω, 2Ω, p(·)). Formally, a lottery over determin-
istic assignments can be thought of as a random variable X that maps each element of
the state space, ω ∈ Ω, into a deterministic assignment. State-of-the-world ω occurs
with probability p(ω), so the probability that a given deterministic assignment x is
drawn is p(x) =∑
ω 1X(ω)=x · p(ω). Hence, X represents a lottery over assignments
in which a given deterministic assignment x is chosen with probability p(x). If two
lotteries can be embedded in some discrete probability space as random variables X
and X ′ in such a way that all agents weakly prefer X(ω) to X ′(ω) for all ω ∈ Ω, with
strict preference for at least one agent in one state ω ∈ Ω, then the lottery embedded
as X is said to state-by-state ex post dominate the lottery embedded as X ′.
Claim 8. x is ex post efficient ⇔ x has a lottery representation that cannot be
state-by-state ex post dominated by any other lottery over feasible deterministic as-
signments
Proof. (⇐) Assume that the support of any lottery representation of ex post efficient
assignment x contains at least one ex post dominated deterministic assignment. Re-
placing this element of the support with its ex post dominator yields the state-by-state
dominator for every lottery representation of x, a contradiction.
(⇒) Assume that any lottery representation of x has a state-by-state dominator.
Then one of these ex post efficient assignments must be ex post dominated, meaning
that there is no representation of x that is a lottery over ex post efficient assignments,
a contradiction.
B Random serial dictatorship with indifferences
Several papers have looked at serial dictatorships adapted for indifferences (Svensson
1994, 1999, Bogomolnaia, Deb and Ehlers 2005), but they have mainly focused on
theoretical characterizations. In this section of the appendix, I will briefly describe
the theoretical mechanism and then two methods by which it can be implemented.
Appendix: p. 1
Let π be a permutation of (1, . . . , |A|), and let ρπ(k) be the rank of aπ(k)’s most
preferred objects that she can be assigned while still assigning agent aπ(k′) something
that he ranked ρπ(k′), for all k′ < k. Define serial dictatorship with indifferences
relative to πππ as the mechanism that yields any deterministic assignment that fulfills
the rank guarantees I just recursively defined.
Basically, this is a random serial dictatorship in which agents are serially as-
signed an indifference class guarantee instead of an actual object. Define random
serial dictatorship with indifferences analogously to the random serial dictator-
ship without indifferences defined in Section 6. Svensson (1994) proves that, even
with indifferences, a deterministic assignment is ex post efficient if and only if it can
be generated by serial dictatorship with indifferences relative to some ordering π.
The extension to random assignments, an analog of Proposition 1 in the main text,
is straightforward.
Proposition 11. An assignment x is ex post efficient ⇔∃ a distribution over dicta-
torship orderings such that the corresponding random serial dictatorship with indif-
ferences can yield x
Also true, and perhaps obvious, is that the extension of random serial dictatorship
to indifferences is strategy-proof.
Proposition 12 (Svensson 1994). Random serial dictatorship with indifferences is
strategy-proof, so long as the distribution over dictatorship ordering is fixed before
agents reveal their preferences.
A natural question at this point is “How does one run this mechanism practically?”
Svensson (1994) offers an algorithm for calculating the opportunity sets of the agents
as the dictatorship progresses, but not for calculating an actual assignment. I offer
two more practical algorithms.
B.1 Integer programming approach
One can easily calculate an agent’s rank guarantee with a simple linear integer pro-
gram
Appendix: p. 2
ρπ(k) = maxx
raπ(k)(o)
s.t.∑
a xao ≤ qo∑o xao = 1
xao ∈ 0, 1raπ(k′)(o) · xaπ(k′)o = ρaπ(k′) · xaπ(k′)o
Any assignment in the arg max of the calculation of ρ|A| is generated by serial dic-
tatorship with indifferences relative to π. The advantage to this approach is that
random serial dictatorship with indifferences can be easily calculated with off-the-
shelf linear optimization software, such as CPLEX or the open-source GNU Linear
Programming Kit (GLPK).
B.2 Augmenting paths approach
The downsides of the integer programming approach are that it uses black box soft-
ware and that for large problems, the integer program might solve quite slowly. For
these reasons, I introduce a more primitive algorithm that can be coded from scratch
if necessary. Consider a bipartite graph connecting agents to objects (qo different
nodes for each object o) with two kinds of links – potential and realized. Realized
links represent tentative allocations, while potential links represent allocations that
could potentially be implemented. At any stage in the algorithm about to be pre-
sented, the realized links will be a near maximum matching in which only one agent
has no realized link, but does have potential links. To see if the unmatched agent can
be matched, look for an augmenting path, which is a path (a1, o2, a3, o4, . . . an−1, on)
where (aj, oj+1) is a potential link, and (oj, aj+1) is a realized link. Implementing
an augmenting path means switching all of its potential links to realized links,
and vice-versa. This necessarily increases the number of agents in the realized link
matching by one. In fact, Berge’s theorem (Berge 1957) tells us that a matching is a
maximum matching if and only if it has no augmenting path. With this background,
the algorithm can be laid out.
1. Initialize.
(a) Create the graph G0
i. Put a potential link between all agents and objects.
ii. Pick any feasible deterministic assignment and put a realized link be-
tween every agent and objects that are assigned to each other.
Appendix: p. 3
(b) Set i = 1 and k = 1.
2. Let Ok be the set of objects in the kth indifference class of agent π(i). Create
Gtempi by deleting all links (implemented and potential) in Gi−1 that connect
to agent π(i) and replacing them with potential links between agent π(i) and
every object in Ok.
3. If there is an augmenting path in Gtempi , implement it, set Gi = Gtemp
i , increment
i, set k = 1, and go to Step 4. If not, then increment k and go to Step 2.
4. If i ≤ |A|, then go to Step 2. Otherwise terminate the algorithm.
When the algorithm terminates, the realized links in G|A| represent a match is gen-
erated by serial dictatorship with indifferences relative to π.
The algorithm is simple. Going through the agents one by one, try to match
them to the best indifference class possible, while preserving a feasible match for the
rest of the agents. Berge’s theorem assures us that if an augmenting path can’t be
found, then our agent in question cannot be matched to anything in the indifference
class being considered. The only computationally difficult parts of this algorithm
are finding the initial matching and searching for an augmenting path in a bipartite
graph. Finding the initial matching can be accomplished via the Hopcroft-Karp
algorithm (Hopcroft and Karp 1973), and finding an augmenting path is typically
accomplished with a breadth first search. Algorithms for these subroutines are easily
found in most textbooks on combinatorial optimization (Schrijver 2003, Cook et al.
1998, Papadimitriou and Steiglitz 1998).
C Simultaneous eating mechanisms and the prob-
abilistic serial mechanism
Bogomolnaia and Moulin (2001) introduced the class of simultaneous eating mecha-
nisms. Essentially, the agents simultaneously claim object shares (“eat”) from their
favorite remaining object, changing objects only when their current favorite is ex-
hausted. Claims occur at a predefined, agent-specific rate which is defined such that
each agent will claim shares that sum to one as time moves from 0 to 1.
Formally, each agent a has a measurable, non-negative eating function ωa :
[0, 1] 7→ R+ such that´ 1
0ωa(t) dt = 1. Let the set of agents who prefer object o the
most out of the set of objects B be given by M(o,B) ≡ a|o a o′,∀o′ ∈ B \ o.
Appendix: p. 4
Simultaneous eating mechanisms are defined by a recursion that will be initialized as
follows:
τ 0 ≡ 0
O0 ≡ O
x0ao ≡ 0, ∀(a, o) ∈ A×O
Every time an object is exhausted, the algorithm moves to a new step of the recursion.
Starting at Step n− 1, let τn(o) be the time at which object o would be exhausted if
the agents in M(o,On−1) were allowed to claim shares of o, uninterrupted, until all
shares of o were gone (setting τn(o) to +∞ for objects that no agent prefers most,
that is, objects such that M (o,On−1) = ∅). Formally,
τn(o) = min
τ∣∣∣∣∣∣
∑a∈M(o,On−1)
ˆ τ
τn−1
ωa(t) dt+∑a
xn−1ao = qo or τ = +∞
Simultaneous eating mechanisms are defined by the following recursion
τn = min
min
o∈On−1τn(o), 1
On = On−1 \ o|τn(o) = τn
xnao =
xn−1ao +
´ τnτn−1 ωa(t) dt if a ∈M(o,On−1)
xn−1ao otherwise
which ends in the first round n′ when τn′= 1. The probabilistic serial mechanism
(PS) is the simultaneous eating mechanism with , ωa(t) = 1 for all agents a and all
t ∈ [0, 1]. As an example, I will derive the outcome under PS from the example in
Figure 1a. Start by calculating the τ 1(o).
Appendix: p. 5
τ 1(a) =1
2
τ 1(b) = 1
τ 1(c) = +∞
τ 1(d) = +∞
Hence, τ 1 = 1/2, O1 = b, c, d, and
x1 =
a b c d
1 : 1/2 0 0 0
2 : 1/2 0 0 0
3 : 0 1/2 0 0
Now, t1(a) =∞, and
τ 1(b) = 3/4
τ 1(c) = 3/2
τ 1(d) = +∞
Hence, τ 2 = 3/4, O2 = c, d, and
x2 =
a b c d
1 : 1/2 0 1/4 0
2 : 1/2 1/4 0 0
3 : 0 3/4 0 0
Finally, τ 2(a) = τ 2(b) =∞, and
τ 1(c) = 9/8
τ 1(d) = 7/4
Hence, τ 2 = 1, and the final probabilistic serial assignment is
Appendix: p. 6
xPS =
a b c d
1 : 1/2 0 1/2 0
2 : 1/2 1/4 0 1/4
3 : 0 3/4 1/4 0
Note that the rank distribution of this assignment is given by NxPS(1) = 7/4;
NxPS(2) = 11/4; NxPS(3) = 3. It is dominated by the assignment x© in Figure 1a.
D The HBS overseas match: indifferences and how
they were broken
Again, most the results from this section are borrowed from Featherstone and Roth
(2011). The HBS match, as mentioned in the main text, allowed for indifferences
to be submitted. It was run using the strategy-proof extension of random serial
dictatorship described in Section B. In the main text, however, I showed results
for strict preferences, which were constructed by breaking the ties in each agent’s
submitted preference. There are two reasonable ways to do this. One is to break
the ties by independently replacing each indifference class with a uniform random
draw from the sets of strict orderings over the objects in the class. Call this the
uncorrelated tie-breaker. The other way I considered was to break all indifferences in
the alphabetical order of the regions, which was also the order in which the regions
were listed on the website used to elicit preferences from the agents. This is called
the correlated tie-breaker. The results from the correlated tie-breaker are presented
in Figure 3 in the main text, while the results from the uncorrelated tie-breaker are
presented in Figure 4 in the Appendix.
The actual results from the HBS match when we deal with indifferences instead of
breaking them are in Figure 5. Also included in the figure is the rank distribution from
the correlated tie-breaker version of uniform random serial dictatorship (U-RSD-T)),
which is the baseline strategy-proof strict preferences baseline that is most commonly
used in assignment markets. Note that accounting for indifferences increases the
percentage of MBAs who get a first choice from 63% to 77%, a more than 20% im-
provement. Though indifferences are not the primary topic of this paper, it is striking
that this sort of improvement can be made without sacrificing strategy-proofness. In-
differences are often treated as a theoretical inconvenience in the matching literature;
the HBS provides evidence that failing to elicit indifferences can leave a significant
Appendix: p. 7
60%
65%
70%
75%
80%
85%
90%
95%
100%
1 2 3 4 5 6 7 8 9 10 11
Ran
ked
kth
or
bet
ter
k
U-RSD
PS
RVM
Figure 4: Results from the uncorrelated tie-breaker(U-RSD is Uniform Random Serial Dictatorship, PS is Probabilistic Serial, and RVM is Rank-Value
Mechanism. Note that U-RSD and PS overlap.)
amount of welfare on the table.
E Proofs
E.1 Rank efficiency
E.1.1 Characterization of rank-value mechanisms
Here, I generalize Theorem 1 to the α-rank efficiency concept defined in Section 9.
Generalization of Theorem 1. x is an α-rank efficient assignment if and only if
there exists a valuation v such that x is supported by the (α, v)-rank-value mechanism.
Proof. Start by proving the “if” part. The objective of the defining linear program
can be rewritten as|O|−1∑k=1
Nxα(k) (vk − vk+1) + |A| · v|O|. By way of contradiction, let x
Appendix: p. 8
60%
65%
70%
75%
80%
85%
90%
95%
100%
1 2 3 4 5 6 7 8 9 10 11
Ran
ked
kth
or
bet
ter
k
PS
RVM
U-RSD
U-RSD-T
Figure 5: Results from the HBS match (no tie-breakers)(U-RSD is Uniform Random Serial Dictatorship, U-RSD-T is Uniform Random Serial Dictatorship
with Tiebreakers, PS is Probabilistic Serial, RVM is Rank-Value Mechanism. Note that U-RSD and
PS overlap.)
Appendix: p. 9
be a random assignment that α-rank-dominates the allocation yielded by the (α, v)-
rank-value mechanism, x∗. Then, the following is true:
|O|−1∑k=1
N xα(k) · (vk − vk+1) + |A| · v|O| >
|O|−1∑k=1
Nx∗
α (k) · (vk − vk+1) + |A| · v|O|
because the α-rank dominance of x over x∗ means that the left-hand side is weakly
greater than the right-hand side, term by term, and strictly so for at least one term.
But this is a contradiction of the optimality of the (α, v)-rank-value mechanism’s
defining linear program.
Now, I prove the “only if” part. Let Aα ⊂ R|O|−1 denote the set of α-rank
distributions that can be generated some random assignment, where the first index
of the vector corresponds to Nxα(1), the second to Nx
α(2), and so on. By the definition
of an α-rank distribution, Aα is convex. Now, let b ∈ Aα be the α-rank distribution
generated by x, the α-rank efficient assignment from the hypothesis of the theorem.
The set of α-rank distributions that would α-rank dominate b if they were generated
by feasible random allocations is U(b) ≡a∣∣a ∈ R|O|−1; ai ≥ bi, ∀i; ∃i′, ai > bi
. U(b)
is also convex. Any member of both Aα and U(b) would α-rank dominate x, so it
must be that Aα ∩ U(b) = ∅. The separating hyperplane theorem then tells us that
there is some p 6= 0 and some c such that p · u ≥ c,∀u ∈ U(b) and p · a ≤ c, ∀a ∈ Aα.
By construction, p · b = c. Without loss of generality, assume c ≥ 0. Now, let ei
denote the unit vector that points along the ith axis of R|O|−1. For any δ>0 , it must
be that b+ δei ∈ U(b). This means that p · (b+ δei) = c+ δpi ≥ c, which means that
pi ≥ 0. Consider the valuation v where vi = 1 +|O|−1∑j=i
pj, ∀i ∈ 1, . . . , |O| − 1, and
v|O| = 1. By construction, the (α, v)-rank-value mechanism supports x. But because
I have yet to prove that pi > 0,∀i, I am not sure that v is a valid valuation, that is
vk > vk+1,∀k.
To show this will require the stronger Polyhedral Separating Hyperplane Theorem
of McLennan (2002). Adopting the notation of that paper’s Theorem 2, let P =
Aα−R|O|−1 (which is a polyhedron by Theorem 1.2 of Ziegler (1995)) and b = p ∈ P .
The α-rank efficiency of x tells us that x is not in the relative interior70 of P , so the
Theorem lets us conclude that there is a hyperplane H such that P is contained in
one of its half-spaces, P ⊂ H−, and F ≡ P ∩H is the smallest face of P that contains
70The relative interior of a set S is its interior in the relative topology of its affine hull. The affinehull of S is the set of all affine combinations of elements of S. An affine combination of a set ofelements s1, . . . , sk of S is a sum
∑i αi · si, where αi ∈ R,∀i and
∑ki=1 αi = 1.
Appendix: p. 10
b. Pick p and c such that H =u ∈ R|O|−1
∣∣p · u = c
. First, note that for δ > 0,
b − δei ∈ P , by definition. Now, say that pi = 0. Then, b − δei ∈ H as well, which
means that b− δei ∈ F . Lemma 2 of McLennan (2002) with S = b tells us that the
relative interior of S, that is b,71 is contained in the relative interior of the smallest
face that contains S, that is F . Hence, for some δ > 0 small enough, b + δei ∈ F .
But since F ⊂ P , this contradicts our original assumption of the α-rank-efficiency of
x. Hence, pi > 0, ∀i.
E.1.2 Rank-improving cycles and improvement cycles
Without changing the number of copies of the objects (including ∅), consider adding∑o qo−|A| dummy agents to the market that are indifferent between all the objects.
Call this the augmented market. Note that the dummy agents are different from
the fictional agent introduced in the definition of claim-trade cycles.72 The extension
of x to the augmented market is an assignment x in which all real agents’ allocations
stay the same and the remaining objects are feasibly assigned to the dummy agents.
Lemma 1. Let x be the extension of x to the augmented market. Then, x is non-
wasteful if and only if there does not exist an improvement cycle in x that includes a
dummy agent.
Proof. (“only if”) If there is an improvement cycle that includes a dummy agent, then
there is also a two agent improvement cycle where one of the agents is the dummy.
Since the dummy is indifferent between all the objects and is holding something that
was not assigned in x, this cycle is equivalent to the real agent claiming something
that was not assigned, contradicting the assumption that x was non-wasteful. (“if”)
If x is wasteful, then x must have a two person improvement cycle between the real
agent who wants to claim an object unassigned under x and the dummy agent who
was assigned that object in x.
Proposition 7 (Bogomolnaia and Moulin 2001). An assignment x is ordinally effi-
cient if and only if x is non-wasteful and admits no improvement cycles.
Proof. This is proven with |A| =∑
o qo for preferences with indifferences in Katta and
Sethuraman (2006), and without indifferences by Bogomolnaia and Moulin (2001).
Ordinal efficiency in a market with |A| >∑
o qo is equivalent to ordinal efficiency in
71The affine hull of b is b.72The fictional agent is, in some sense, a “super” dummy that owns the shares held by all of the
smaller dummies.
Appendix: p. 11
the augmented market, so ordinal efficiency is equivalent to no cycles in the augmented
market. A cycle in the augmented market either contains a dummy or doesn’t, so
by Lemma 1, a cycle in the augmented market is equivalent to either a cycle or
wastefulness in the regular market.
Before addressing α-rank efficiency, I re-introduce the generalization of a v-rank-
improving cycle that was relegated to Footnote 62 in the main text.
Definition 16. A trade cycle τ on assignment x is an (α, v)-rank-improving cycle
if
|τ |∑k=1
αak ·[vrak (ok−1) − vrak (ok)
]> 0 (with o0 ≡ o|τ |).
Lemma 2. Let x be the extension of x to the augmented market. Then, x admits no
(α, v)-rank improving cycles if and only if x admits no (α, v)-rank improving cycles.
Remark 8. Note that while there can claim-trade cycles in x, there can’t in x, since
by construction, all object shares are claimed.
Proof. (“only if”) By way of contradiction, assume that τ is an (α, v)-rank improving
cycle in x. If τ has exactly one dummy, then it is already an (α, v)-rank improving
claim-trade cycle in x, a contradiction. If τ has multiple dummies, then it can be par-
titioned into several claim-trade cycles in x by taking the parts between any dummy
and the next dummy in τ . Because τ is overall (α, v)-rank improving, at least one
of the elements of the partition must be an (α, v)-rank improving claim-trade cycle
in x, a contradiction. Obviously, if τ has no dummies, then it is also an (α, v)-rank
improving trade cycle in x, a contradiction. (“if”) By way of contradiction, assume
that τ is an (α, v)-rank improving claim-trade cycle in x. Then an (α, v)-rank im-
proving trade cycle can be constructed in x by letting a1 be a dummy that is holding
o1, a contradiction. Now, assume that τ is an (α, v)-rank improving trade cycle in x.
It must also be an (α, v)-rank improving trade cycle in x, a contradiction.
Generalization of Theorem 2. x is α-rank efficient if and only if there exists a
valuation v such that x admits no (α, v)-rank-improving cycles.
Proof. (“only if”) By way of contradiction, assume that for any valuation v, there
exists an (α, v)-rank improving cycle. This would yield an improvement in the ob-
jective of the (α, v)-rank-value mechanism, which means that x cannot be generated
by any (α, v)-rank-value mechanism, which, by the Generalization of Theorem 1 in
Section E.1.1 of the Appendix, means that x is not α-rank efficient.
Appendix: p. 12
(“if”) Consider the extension of x to the augmented market, x. Now, by way of
contradiction, let x α-rank dominate x in that market. Let ξ ≡ x − x. Necessarily,∑o ξao = 0 and
∑a ξao = 0. Construct a cycle by the following procedure. First,
find some (a1, o1) such that ξa1o1 < 0. Then, find some a2 such that ξa2o1 > 0. Next,
find some o2 such that ξa2o2 < 0. In words, this process starts by picking a negative
element in ξ, then goes down the column to a positive element, then across the row
to a negative element, and so on. Continue building a sequence of agents and objects
in this way, until an agent is visited twice, i.e., a cycle is found. This process is
well-defined because any row or column with a non-zero element must have another
non-zero element of the opposite sign, since all rows and columns sum to zero. The
process will eventually terminate because the market is finite. By construction, the
process finds sequence of agents and objects that is a trade cycle. Denote it by τ .
Now, let Ξ = min(a,o)∈τ
|ξao| and define:
ξ1ao =
Ξ ∃k s.t. (a, o) = (ak, ok)
−Ξ ∃k s.t. (a, o) = (ak+1, ok)
0 otherwise
Then ξ− ξ1 has at least one fewer non-zero entries than ξ. Hence, this process can be
repeated to decompose ξ into a sum of trade cycles, that is ξ =∑
i ξi where each ξi
represents a trade cycle in the augmented market. Then, x = x +∑
i ξi. Now, since
x α-rank dominates, the objective of the (α, v)-rank-value mechanism is bigger at x,
regardless of the valuation. By the linearity of the objective of the (α, v)-rank-value
mechanism, it cannot be that the value of the objective evaluated at ξi is weakly
negative for all i. In other words, regardless of the valuation, at least one of the ξis
must be (α, v)-rank improving, which by Lemma 2 means that there is an (α, v)-rank
improving cycle in x, which by Lemma 2 is equivalent to an (α, v)-rank improving
cycle in x, a contradiction.
Proposition 4. If all elements of the initial assignment matrix are rational numbers,
then the sequential improvement process will terminate in a finite number of steps with
a rank efficient assignment that is in the arg max of the v-rank-value mechanism.
Proof. Implementing an (α, v)-rank-improving cycle transforms deterministic assign-
ments into deterministic assignments. It also necessarily increases the objective of
the (α, v)-rank-value mechanism, so the algorithm cannot cycle. Since the set of de-
Appendix: p. 13
terministic assignments is finite, the algorithm must terminate. Note that the same
logic works if all entries in the initial matrix are rational: simply multiply the starting
matrix by the least common denominator.
Lemma 3. Let x be a deterministic assignment. Then x is ex post efficient if and
only if it is non-wasteful and admits no improvement cycles.
Proof. The “only if” follows from the fact that implementing any improvement cycle
in the augmented market immediately yields an ex post dominator. For the “if” part,
consider the extension of x to the augmented market, x and assume it has an ex post
dominator, x. Using the decomposition strategy of the proof to the Generalization of
Theorem 2, one can move from x to x through a series of trade cycles. But, because
both x and x are deterministic assignments, for all the trade cycles, Ξ will equal 1.
Hence, each agent can only be involved in, at most, one of the trade cycles, and hence,
all of the trade cycles must be improvement cycles. By Lemma 1, these cycles map
back to either an improvement cycle, or wastefulness in x, a contradiction.
Proposition 8. Let x be a deterministic assignment. Then,
(i) x is ordinally efficient if and only if it is ex post efficient.
(ii) x is rank efficient implies it is ex post efficient; however the converse need not
hold.
Proof. Part (i) follows directly from Proposition 7 and Lemma 3. Now, I prove (ii).
Proposition 3 establishes that rank efficiency implies ex post efficiency. To see that
the converse need not hold, consider x© and y from Figure 1a. Both are ex post
efficient deterministic assignments, but only one is rank efficient.
E.1.3 Ex ante efficiency interpretation
The proof in the main text only addresses uniform weights. To justify an (α, v)-
rank-value mechanism, I need αa ·´va′k · F (v) = αa′ ·
´vak · F (v). In other words,
the unconditional expectations of va for all agents should be proportional, and the
constants of proportionality should be the appropriate weights from the α vector.
E.2 Truth-telling in low information environments
I allow for indifferences and a general weighting of the agents in all theorems I will
be deriving, so when I generalize relative to a statement in the main text, I will
Appendix: p. 14
note that I have done so . The general scheme of the proofs in this section borrow
heavily from Roth and Rothblum (1999), but the proofs themselves are new, as they
must be adapted to the rank-value mechanism, which is quite different from the more
algorithmically defined deferred acceptance algorithm. Let RVMα,va (%; q; τ) denote
a’s allocation under the (α, v)-rank-value mechanism when the submitted preferences
are %, the capacities are q, and the tie-breaker ordering is τ . Let the objective
function of the (α, v)-rank-value mechanism evaluated at assignment x and submitted
preferences % be denoted by
V α,v(%;x) ≡∑a′′
∑o′′
αa′′ · vra′′ (o′′) · xa′′o′′
Lemma 4. Let o′ a o. Then,
[RVMα,va (%; q; τ) = o]⇒
[RVMα,v
a
(%o↔o′a ,%−a; q; τ
)= o]
Proof. Swapping o and o′ changes the objective in the following way:
V α,v(%o↔o′a ,%−a;x
)= V α,v(%;x) + αa ·
(vra(o′) − vra(o)
)· (xao − xao′) (1)
Now, let y = RVMα,v (%; q; τ), which tells us that yao = 1 and yao′ = 0, which means
V α,v(%; y)− V α,v(%o↔o′a ,%−a; y
)= −αa ·
(vra(o′) − vra(o)
)< 0
Now consider another assignment z 6= y that is a member of the arg max of the linear
program when when o and o′ are switched, that is
V α,v(%o↔o′a ,%−a; z
)≥ V α,v
(%o↔o′a ,%−a; y
)(2)
First, I show that zao = 1. Assume otherwise. Then, by Equation 1, I derive
V α,v (%; z)− V α,v(%o↔o′a ,%−a; z
)≥ 0 > V α,v (%; y)− V α,v
(%o↔o′a ,%−a; y
)If I add this inequality to Inequality 2, I find that y was not an optimum to start
with, a contradiction. Hence, a must receive o when he switches. Now, if zao = 1,
then following the same logic, I derive
V α,v (%; z)− V α,v(%o↔o′a ,%−a; z
)= V α,v (%; y)− V α,v
(%o↔o′a ,%−a; y
)
Appendix: p. 15
and combine it with Inequality 2 to find that either y was not an optimum to start
with (a contradiction), or that z is a member of the arg max under both normal and
switched preferences. But then, the only way that z could be chosen under the switch
is if τ(z) < τ(y), which would contradict y being chosen under the truth.
Corollary (to Lemma 4). Let o′ a o. Then,[RVMα,v
a
(%o↔o′a ,%−a; τ
)= o′
]⇒ [RVMα,v
a (%; τ) = o′]
Proposition 9. Under a rank-value mechanism, if agent a’s beliefs are o, o′-symmetric,
and o′ a o, then for a, the allocation resulting from any submitted preference that
declares o a o′ is weakly stochastically dominated by the allocation resulting from a
submitted preference that does not. If the beliefs are symmetric, then this holds for
all o, o′ ∈ O \ ∅.
Proof. First, note that since the linear program does not depend on labels,
[RVMα,va (%; q; τ) = v]⇔
[RVMα,v
a
(%o↔o′ ; qo↔o
′; τ o↔o
′)
= v]
[RVMα,va (%; q; τ) = o]⇔
[RVMα,v
a
(%o↔o′ ; qo↔o
′; τ o↔o
′)
= o′]
where o, o′ ∈ O and v ∈ O \ o, o′. Then, it must also be that
[RVMα,v
a
(o↔o′a ,%−a; q; τ
)= v]⇔[RVMα,v
a
(%a,%
o↔o′−a ; qo↔o
′; τ o↔o
′)
= v]
[RVMα,v
a
(%o↔o′a ,%−a; q; τ
)= o]⇔[RVMα,v
a
(%a,o↔o
′
−a ; qo↔o′; τ o↔o
′)
= o′]
Now, using Lemma 4 and its corollary, one can think about the gains and losses an
agent a who prefers o′ to o gets from switching those two objects. The possible cases
are shown in Table 2. Using these cases, one can then think about what our agent
gets when everyone else’s preferences switch o and o′, which is equally likely due to
the assumption of o, o′-symmetric beliefs. This analysis is shown in Table 3. In
each of the cases, considering that everyone else switching o and o′ is equally likely,
truth-telling yields an allocation for a that weakly stochastically dominates what she
would have gotten by switching o and o′.
The condition in Proposition 9 is important enough to warrant a name.
Appendix: p. 16
Lie: RVMα,va
(%o↔o′a ,%−a; q; τ
)= u /∈o, o′ = o′ = o
= v /∈ o, o′ Case A Impossible Case B
Truth: RVMa (%a,%−a; q; τ) = o′ Case C Case D Case E
= o Impossible Impossible Case F
Table 2: Cases
a tells the truth a switches o and o′
(%a,%−a; q; τ)(%a,%o↔o
′−a ; qo↔o
′; τo↔o
′) (
%o↔o′
a ,%−a; q; τ) (
%o↔o′
a ,%o↔o′
−a ; qo↔o′; τo↔o
′)
Case A v u u v
Case B v o′ o v
Case C o′ u u o
Case D o′ o o′ o
Case E o′ o′ o o
Case F o o′ o o′
Table 3: Assignments under the different cases
Definition 17. A reported preference %′a is called weakly o, o′o, o′o, o′-order-preserving
with respect to %a%a%a if [o′ a o]⇒ [o′ %′a o]. If this is true for all o, o′ ∈ O\∅, then
just call the reported preference weakly order-preserving with respect to %a%a%a.
Notice that if I am dealing with a world where only strict preferences are allowed,
then this property basically says that objects besides ∅ are listed in their true order,
which is where I get Theorem 4. If I allow for indifferences, then I am not insisting on
truthful ordering, but instead are insisting that a true strict ordering of two objects is
never reported reversed. Before moving on, I review a notation that was only briefly
introduced in the main text. Let the true preference be %a and the preference that
adds an indifference class I of objects right above ∅ be %+Ia . For some object o, let
%+oa denote %+o
a .
Lemma 5. If the ranking scheme is upward-looking and unmatched-distinct then,
submitting a preference where the bottom declared-acceptable indifference class con-
sists entirely of truly unacceptable objects is weakly stochastically dominated by the
submitting the same preference with the bottom declared-acceptable indifference class
dropped.
Appendix: p. 17
Proof. By way of contradiction, assume that I is a set of unacceptable objects,
and that submitting %+Ia makes a better off, that is RVMa
(%+Ia ,%−a; q; τ
)≡ z,
RVMa (%; q; τ) ≡ y, and ya ≺a za. Then, by optimality, I know that V (%; y) ≥V (%; z) and V
(%+Ia ,%−a; z
)≥ V
(%+Ia ,%−a; y
). Now, ya is necessarily not in the
lowest indifference class of %+Ia , but it could either be in a higher indifference class,
or it could be ∅.
Case 1. (ya is ranked above %+Ia ) za must also be ranked above the bottom accept-
able indifference class of %+Ia , so (since I have assumed that the rankings
are upward-looking) V(%+Ia ,%−a; y
)= V (%; y) and V
(%+Ia ,%−a; z
)=
V (%; z). Combined with the optimality inequalities, these equalities im-
ply that y and z are both in the arg max under both preferences. This
cannot be, since the tiebreaker τ would choose one or the other.
Case 2. (ya = ∅) za is necessarily not in the lowest acceptable indifference class
of %+Ia , so V
(%+Ia ,%−a; z
)= V (%; z). Now since the ranking scheme
is unmatched-distinct, V(%+Ia ,%−a; y
)= V (%; y). Stringing inequalities
together, I find that y and z are both in the arg max under both preferences.
This cannot be, since the tiebreaker τ would choose one or the other.
Lemma 6. If the ranking scheme is upward-looking and unmatched-neutral, then if
o is a truly acceptable object that is declared unacceptable in %a, then
RVMα,va (%; q; τ) -a RVM
α,va
(%+oa ,%−a; q; τ
)Proof. Define y ≡ RVMα,v (%; q; τ) and z ≡ RVMα,v (%+o
a ,%−a; q; τ). Since y is an
arg max under %, I have V α,v (%; y) ≥ V α,v(%; z), and since z is an arg max under
(%+oa ,%−a), I have V α,v (%+o
a ,%−a; z) ≥ V α,v(%+oa ,%−a; y).
Now, by way of contradiction, assume that ya a za. There are three cases.
Case 1. (za = ∅) The ranking scheme is unmatched-neutral, so V α,v(%; z) > V α,v(%+oa
,%−a; z). Since ya is above the part of the rank-order that has changed, I
know V α,v(%+oa ,%−a; y) = V α,v(%; y) by the upward-looking assumption.
Combined with the optimality inequalities, I quickly arrive at a contradic-
tion.
Case 2. (za = o) Denote by z|(a,∅) the assignment I get by starting with z and only
moving a from o to ∅. This is feasible, so since y is an arg max under %, I
Appendix: p. 18
know that V α,v (%; y) ≥ V α,v(%; z|(a,∅)) By unmatched-neutrality, V α,v(%
; z|(a,∅)) = V α,v(%+oa ,%−a; z) Again, since ya is above any changes to the
rank orders, V α,v(%; y) = V α,v(%+oa ,%−a; y). Combined with the fact that
z is an arg max under (%+oa ,%−a), I can deduce that all of these objective
values must be equal. But if this were true, there would no way that the
same tiebreaker τ could have chosen two different assignments.
Case 3. (za a o) Both za and ya are above any changes in the rank orders. Hence,
by the upward-looking assumption, z and y must be in the arg max for
both submitted preferences, and since I break ties by choosing the lowest
τ , it is a contradiction that I choose y under the true preferences and z
under the truncations.
Proposition 13. Assume that agent a’s beliefs are symmetric. If the ranking scheme
is upward-looking and unmatched-neutral, then truly acceptable objects must be de-
clared acceptable. If the ranking scheme is upward-looking and unmatched-distinct,
then any unacceptable objects that a declares acceptable must be in the bottom indif-
ference class and declared indifferent to some acceptable objects.
Proof. The previous lemma establishes the first part of the theorem. For the second
part, note that symmetry requires that the objects are submitted in a weakly order
preserving way. Hence, there can exist only one indifference class that contains both
unacceptable and acceptable objects. Below that indifference class, all classes must
be filled with unacceptable objects. By Lemma 5, I can drop them.
This combined with Proposition 9 gives us the generalization of Theorem 5 to
preferences with indifferences.
Generalization of Theorem 5. If an agent’s beliefs are symmetric and the rank
scheme is upward-looking and...
• ...unmatched-distinct, then his expected allocation from any other strategy is
weakly stochastically dominated by his expected allocation from submitting some
weakly order preserving preference in which any truly unacceptable objects that
are declared acceptable are declared indifferent to some truly acceptable object
and are declared in the lowest acceptable indifference class.
Appendix: p. 19
• ...unmatched-neutral, then his expected allocation from any other strategy is
weakly stochastically dominated by his expected allocation from submitting some
weakly order preserving preference in which all truly acceptable objects are de-
clared acceptable.
E.3 Supportable rank efficiency
Theorem 6. Supportable rank efficiency implies ordinal efficiency. The converse
need not hold.
Proof. To show the positive statement, note that supportable rank efficiency im-
plies, for some α, α-rank efficiency, which in turn, for any α, implies ordinal effi-
ciency. To see that the converse need not hold, start by defining ∆(α,v)τ V =
|τ |∑m=1
αam ·(vram (om−1)−vram (om)
)for a trade cycle τ (where o0 ≡ om). By the Generalization of
Theorem 2 in Section E.1.2, an assignment is only supportably rank efficient there
exists some (α, v) such that ∆(α,v)τ V ≤ 0 for all trade cycles τ . Now, consider a 6
agent example, where there is one seat at a through d and 2 seats at e.
1 : a d e
2 : a c e
3 : a b e
4 : b c e
5 : c d e
6 : b e
Probabilistic serial yields the random allocation
a b c d e
1 : 1/3 0 0 17/27 1/27
2 : 1/3 0 7/27 0 11/27
3 : 1/3 1/9 0 0 15/27
4 : 0 4/9 4/27 0 11/27
5 : 0 0 16/27 10/27 1/27
6 : 0 4/9 0 0 15/27
which I know to be ordinally efficient. Relative to the trade cycles
Appendix: p. 20
τ1 = ((1, a) , (2, e))
τ2 = ((1, e) , (2, a))
τ3 = ((1, a) , (3, b) , (4, c) , (5, d))
τ4 = ((1, a) , (2, e) , (3, b) , (4, c) , (5, d))
I can calculate how ∆(α,v)τ V will change:
∆(α,v)τ1 V = (α2 − α1) · (v1 − v3)
∆(α,v)τ2 V = (α1 − α2) · (v1 − v3)
∆(α,v)τ3 V = (α3 + α4 + α5 − α1) · (v1 − v2)
∆(α,v)τ4 V = −α1 · (v1 − v2) + α2 · (v1 − v3)− α3 · (v2 − v3) + (α4 + α5) · (v1 − v2)
Now, I show how there is no (α, v) such that one of these trade cycles isn’t (α, v)-
rank improving, that is ∆(α,v)τ V ≤ 0. Since v1 > v3, I need α1 = α2 to prevent
either τ1 and τ2 from being (α, v)-rank-improving. To prevent τ3 from being α-rank-
improving, I need α1 ≥ α3 +α4 +α5. Finally, since α1 = α2, I can simplify ∆(α,v)τ4 V to
(α1−α3)·(v2−v3)+(α4+α5)·(v1−v2). The second term is definitely positive, so it must
be that α1 < α3. But combining the last two inequalities gives us α3 > α3 + α4 + α5,
which cannot be. Hence, there is no (α, v) such that ∆(α,v)τ V ≤ 0 for all four of the
trade cycles I have listed. Thus the ordinally efficient assignment I started with is
not supportably rank efficient.
E.4 Competitive equilibrium mechanisms
To establish Theorem 7, I will be using the concept of a discount equilibrium.
Definition 18. Prices and an assignment, (x∗, p), form a discount equilibrium with
respect to weights ααα and valuation vvv (or just (α,v)-discount equilibrium) if, for
all a,
x∗a ∈ arg maxxa≥0
∑o
(vra(o) − po
αa
)· xao
s.t.∑o
xao ≤ 1
and
Appendix: p. 21
x∗ao ∈ arg maxxao≥0
∑a
∑o
po · xao
s.t.∑a
xao ≤ qo ,∀o
(α, v)-discount equilibria decentralize the linear program that defined the (α, v)-
rank-value mechanism. Formally,
Proposition 14. x is α-rank efficient if and only if there exist prices p, and a valu-
ation v, such that (x∗, p) form an (α, v)-discount equilibrium
Proof. First, I prove the “only if” part. Since x is α-rank efficient, there must be a
valuation such that it is in the arg max of the (α, v)-rank-value mechanism, by the
Generalization of Theorem 1 in Section E.1.1.
The LP dual of the (α, v)-rank efficient mechanism is
minua≥0,po≥0
∑o∈O
qo · po +∑a∈A
αa · ua
s.t. ua ≥ vra(o) − poαa
∀o ∈ O, a ∈ A
I will support the discount equilibrium with the prices from the dual of the rank
efficient mechanism. Now, the LP dual of the agent a’s problem in the (α, v)-discount
equilibrium with these prices is
minua
ua
s.t. ua ≥ vra(o) − poαa∀o ∈ O
By nested optimization, I see that the ua from the rank efficient mechanism’s
dual also solves the discount equilibrium duals. Now, let xao be the assignment I
are considering from the arg max of the rank efficient mechanism. The LP duality
theorem tells us that
∑o∈O
qo · po +∑a∈A
αa · ua =∑a∈A
∑o∈O
αa · vra(o) · xao
From the constraint in the dual of the rank efficient mechanism, I can then derive
that, for any ξ such that∑
o ξao ≤ 1,∀a ∈ A, I have
∑o∈O
qo · po +∑a∈A
αa ·∑o∈O
(vra(o) −
poαa
)· ξao ≤
∑a∈A
∑o∈O
αa · vra(o) · xao (3)
If I plug in x for ξ, then this reduces to∑
o∈O po ·(qo −
∑a∈A xao
)≤ 0. Since
all elements of that sum are weakly positive, I conclude that po ·(qo −
∑a∈A xao
)=
Appendix: p. 22
0,∀o ∈ O. This, coupled with a constraint from the rank efficient mechanism, is the
second condition of the discount equilibrium. Note that I have also shown that
∑o∈O
qo · po +∑a∈A
αa ·∑o∈O
(vra(o) −
poαa
)· xao =
∑a∈A
∑o∈O
αa · vra(o) · xao
Finally, I show that the xao from the arg max of the rank efficient mechanism must
also be part of the arg max for the agents’ problems in the discount equilibrium.
By way of contradiction, say that they aren’t, and let ξa be the bundle chosen by
agent a under prices p. Then, for all agents,∑
o ξao ≤ 1 and∑
o
(vra(o) − po
αa
)· ξao ≥∑
o
(vra(o) − po
αa
)· xao, with the second inequality strict for some agent a′. Thus,
∑o∈O
qo ·po+∑a∈A
αa ·∑o
(vra(o) −
poαa
)·ξao >
∑o∈O
qo ·po+∑a∈A
αa ·∑o
(vra(o) −
poαa
)·xao
But I have shown∑o∈O
qo · po +∑a∈A
αa ·∑o
(vra(o) −
poαa
)· xao =
∑a∈A
∑o∈O
αa · vra(o) · xao,
which means that I have derived∑o∈O
qo · po +∑a∈A
αa ·∑o
(vra(o) −
poαa
)· ξao >
∑a∈A
∑o∈O
αa · vra(o) · xao
which stands in direct contradiction to something I have already proven, Inequality 3.
Therefore, the assignment from the rank efficient mechanism solves the agents’ prob-
lems in the discount equilibrium. Thus I have shown that any member of the arg
max of the rank efficient equilibrium can be supported as a discount equilibrium.
Now, I prove the “if” part. Consider the optimization
maxx
∑o
∑a
αa · vra(o) · xao +∑o
po ·(qo −
∑a
xao
)s.t.
∑o
xao ≤ 1∑a
xao ≤ qo
By the positivity of the prices and the object quota constraints, the value of this
optimization is greater than or equal to the value of
Appendix: p. 23
maxx
∑o
∑a
αa · vra(o) · xao
s.t.∑o
xao ≤ 1∑a
xao ≤ qo
Now, if I start with the assignment x from some discount equilibrium, it will also
solve the first optimization. To see this, relax the optimization by removing the object
capacity constraints. The relaxation can then be separated into an optimization for
each agent that is equivalent to the agent’s problem from the discount equilibrium,
so x solves the relaxation. But, since x is part of a discount equilibrium, it also meets
the object capacity constraints, and is thus an optimizer of the full problem as well.
By the seller’s problem in the discount equilibrium (see Remark 7), the value of this
assignment in the first optimization is attained in the second as well. Hence, x must
solve the second optimization, since it attains the upper bound at x. Hence, x is
in the arg max of the (α, v)-rank-value mechanism’s defining program, which means
that x is α-rank efficient.
This is a strange decentralization, as it distorts prices in a way that one might
expect it to not yield any sort of efficiency. At the end of this section, I will discuss
the intuition for why the discount equilibrium decentralization remains efficient.
Proposition 15. There exist weights α such that (x, p) is an (α, v)-discount equilib-
rium if and only if there exist budgets B such that (x, p) is a (B, v)-budget equilibrium.
Proof. I start by showing that if I start with an (α, v)-discount equilibrium (x∗, p),
then it is also a (B, v)-budget equilibrium for Ba =∑o
po · x∗ao. By way of con-
tradiction, assume that (x∗, p) is not a (B, v)-budget equilibrium assignment for
B. By construction, it is feasible, so it must be that there is some other feasi-
ble x such that∑o
vra(o) · xao >∑o
vra(o) · x∗ao. Since it is feasible, I also know that∑o
po · xao ≤∑o
po · x∗ao. Combining the last two inequalities (multiplying the second
by 1/αa), I find∑o
(vra(o) − po
αa
)· xao >
∑o
(vra(o) − po
αa
)· x∗ao, a contradiction of the
original assumption that (x∗, p) was an (α, v)-discount equilibrium.
Now, I show that if I start with a (B, v)-budget equilibrium, (x, p), then it is also
an (α, v)-discount equilibrium for some choice of α. Before I define our weights, how-
ever, I first introduce a few LP duals. The dual of the agent optimization problem in
Appendix: p. 24
the budget equilibrium is(λa, ua
)∈ arg min
λa,ua≥0ua + λa ·Ba
s.t. ua ≥ vra(o) − po · λa,∀a
The LP dual of the agent optimization problem in an (α, v)-discount equilibrium is
u∗a ∈ arg minua≥0
ua
s.t. ua ≥ vra(o) − poαa
,∀a
I now show that each agent a can be made to optimize by choosing xa under the
discount equilibrium for some choice of αa. Tildes indicate optimal choices from the
budget equilibrium that I start with. From here I continue by cases.
Case 1. (λa = 0): By inspection of the dual, I can see that ua = v1. By the LP
duality theorem, I then know that∑
o vra(o) · xao = ua = v1. The only
way this can happen is if xao = 1ra(o)=1, that is, if agent a is able to
buy a full share of his most preferred object, r−1a (1). without exhausting
his budget. Clearly, if αa → ∞, then a will also take a full share of his
most preferred object in the discount equilibrium, as the influence of prices
vanishes in the limit. This can be done with a finite αa as well: simply
set αa high enough that vk −pr−1a (k)
αais a maximum at k = 1, that is set
αa = maxk
pr−1a (1)
−pr−1a (k)
v1−vk
. I know this is finite because v1 > vk,∀k and
prices must be finite in a budget equilibrium.73
Case 2. (λa > 0): Set αa = 1
λaand let stars indicate optimal choices from agent a’s
problem in the corresponding discount equilibrium. Our choice of αa nests
our two LP duals, so that u∗a = ua. The LP duality theorem then gives us
that∑
o
(vra(o) − po
αa
)·x∗ao = u∗a = ua =
∑o vra(o)·xao−Ba
αa. Feasibility of our
starting budget equilibrium requires that∑
o vra(o) · xao ≤ Ba, which, when
combined with our duality equality, tells us that∑
o
(vra(o) − po
αa
)· x∗ao ≤∑
o
(vra(o) − po
αa
)· xao. A strict inequality here contradicts x∗a being agent
a’s optimum under the discount equilibrium. Hence, xa is an optimum
choice for a under the discount equilibrium with the αa I have chosen.
Hence I have shown that agents optimizing according to the problem in the discount
equilibrium can choose x, given an appropriately chosen α vector.
73If there were an infinite price, no agent could purchase the object, since budgets are finite, thiswould necessarily violate the po > 0⇒
∑a xao = qo requirement.
Appendix: p. 25
Theorem 7. x is supportably rank efficient if and only if there exist budgets and a
valuation, (B, v), such that x is supported by the (B, v)-budget mechanism.
Proof. The previous two proofs establish the result.
The efficiency of the budget equilibria lines up well with a textbook understanding
of general equilibrium and the welfare theorems, but it is less clear why the discount
equilibrium should yield an efficient outcome, as distorting prices with the α-discounts
seems like it should cause trouble in the same way that distortionary taxation does.
So what is going on? The agent’s problem in the discount equilibrium can be thought
of as the problem of an agent with enough money to buy whatever he wants, but who
values money quasi-linearly. In the artificial economy where money factors into the
agent’s utility, the discount equilibrium will not be efficient. The efficiency concepts
in this paper, however, ignore the artificial money and rely only on the distribution
of objects among the agents. In short, in the discount equilibrium, distortions in
the relative cost of keeping money serve a similar role to the budgets in the budget
equilibrium, but they do so in a way that allows more precise control over what Pareto
weights will support the chosen optimal assignment (at the cost of sacrificing control
over agents’ opportunity sets).
Appendix: p. 26